AAdditive C ∗ -categories and K -theory Ulrich Bunke ∗ Alexander Engel † October 29, 2020
Abstract
We introduce and study the notion of an orthogonal sum of a (possibly infinite)family of objects in a C ∗ -category. Furthermore, we construct reduced crossedproducts of C ∗ -categories with groups. We axiomatize the basic properties of the K -theory for C ∗ -categories in the notion of a homological functor. We then studyvarious rigidity properties of homological functors in general, and special additionalfeatures of the K -theory of C ∗ -categories. As an application we construct and studyinteresting functors on the orbit category of a group from C ∗ -categorical data. Contents C -linear ∗ -categories and C ∗ -categories 63 Orthogonal sums in C ∗ -categories 114 Morphisms into and out of orthogonal sums 245 Multiplier categories and Antoun–Voigt sums 326 Isometric embeddings of C ∗ -categories and orthogonal sums 417 The category of α -additive C ∗ -categories 468 Non-existence of an additive completion functor 50 ∗ Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, [email protected] † Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, GERMANYa [email protected] a r X i v : . [ m a t h . K T ] O c t Orthogonal sums of functors and weakly G -invariant functors 5510 Reduced crossed products 5811 Homological functors 7012 Topological K -theory of C ∗ -categories 7613 K -theory of products of C ∗ -categories 8214 Morita invariance 9115 Relative equivalences and idempotent completions 10116 Weak Morita equivalences 10717 Functors on the orbit category 114 References 135
The goal of this paper is to provide a reference for foundational results about C ∗ -categoriesand their topological K -theory. The three main themes are orthogonal sums of (infinite)families of objects in a C ∗ -category, reduced crossed products of C ∗ -categories with groups,and rigidity properties of the K -theory of C ∗ -categories and more general homologicalfunctors which go beyond the invariance under unitary equivalences. The results of thepresent paper will be used in the subsequent papers [BE] and [BEL].The notion of a C ∗ -category was introduced in [GLR85]; see also the further references[Del12, DL98, Joa03, Mit02, Mit04, Bun19]. The category C ∗ Cat of C ∗ -categories has aninteresting homotopy theory based on the notion of unitary equivalence which is studiedin [Del12, Bun19].The main topic of [Bun] are the categorical properties of the category C ∗ Cat nu of possiblynon-unital C ∗ -categories. In particular it was shown that this category is complete andcocomplete. Furthermore, for C ∗ -categories with G -action the maximal crossed productwas introduced and recognized as a homotopy colimit.One aim of [BE] is to construct equivariant coarse homology theories in the sense of[BE20, BEKW20a] associated to a coefficient C ∗ -category following the recipe of [BE20,BEKW20a, BCKW]. In a first step we functorially associate to every bornological coarsespace a C ∗ -category of controlled objects in the coefficient category. We then apply a2omological functor from C ∗ -categories to some target category. In order to verify that thisconstruction satisfies the axioms of an equivariant coarse homology theory we first verifyrelated properties of these controlled object categories and then use the defining propertiesof the homological functor. While [BE] focusses on the controlled object functors, thepurpose of this paper is to provide the necessary background about C ∗ -categories andhomological functors.In [BEL] we construct a stable ∞ -category KK G modelling equivariant Kasparov KK -theory and derive an equivariant version of Paschke duality which is in turn used tocompare the analytic and homotopy theoretic versions of the Baum–Connes assembly map.Again [BEL] uses orthogonal sums, crossed products and various properties of K -theoryshown in the present paper.In the remainder of this introduction we describe the content of this paper in greaterdetail.Section 2 serves as a reminder of basic notions from the theory of C ∗ -categories.The first main topic of the present paper are orthogonal sums of families of objects in a C ∗ -category. We have various reasons for considering such sums:1. Let X be a set. The main feature of the definition of an X -controlled object C ina C ∗ -category is a presentation of C as an orthogonal sum of a family of objects( C x ) x ∈ X indexed by the set X . See Definition 17.7 for an instance of this idea.2. In Definition 10.8 the reduced crossed product of a C ∗ -category ˜ C with G -action willbe constructed by completing the algebraic crossed product (introduced in [Bun])with respect to a norm obtained from a representation on a C ∗ category derivedfrom C which we will denote suggestively by L ( G, ˜ C ). In particular, the morphismspaces of the latter are given, using orthogonal sums of families of objects indexedby G , in terms of the morphism spaces of C by Hom C ( (cid:76) g ∈ G gC, (cid:76) g ∈ G gC (cid:48) ).3. Frequently the fact that a C ∗ -category C has trivial K -theory is deduced from anEilenberg swindle. This will be encoded in the notion of flasqueness of C , see theDefinition 9.3. The usual verification of flasqueness of C consists in showing that forevery object C the infinite sum (cid:76) N C of countably many copies of C exists in C .If A is a C ∗ -algebra, then the category Hilb ( A ) of A -Hilbert C ∗ -modules is an exampleof a C ∗ -category. Given some family ( M i ) i ∈ I of objects in Hilb ( A ) we can constructthe orthogonal sum (cid:76) i ∈ I M i in Hilb ( A ) as a completion of the algebraic direct sumwith respect to the norm induced by an explicitly given A -valued scalar product. Wecan then understand the spaces of bounded adjointable operators B ( (cid:76) i ∈ I M i , M ) or B ( M, (cid:76) i ∈ I M i ) for any object M in Hilb ( A ). For general C ∗ -categories our idea is tofind characterizations of the spaces B ( (cid:76) i ∈ I M i , M ) and B ( M, (cid:76) i ∈ I M i ) without usingthe construction of (cid:76) i ∈ I M i . The details are inspired by the abstract construction of3ultiplier algebras. Our final definition of an orthogonal sum of a family of objects in aunital C ∗ -category is Definition 3.16. In Lemma 3.23 we show that in the case of Hilb ( A )our definition reproduces the classical definition. Section 4 provides additional materialwhich is helpful when working with sums.In Section 5 we review in detail the notion (see Definition 5.7) of orthogonal sums (calledAV-sums) due to Antoun and Voigt [AV] . The notion of an AV-sum is based on multipliercategories and differs from our notion. In order to produce a detailed reference we first recallthe construction of multiplier categories. In Proposition 5.9 we then give a precise way ofcomparing the two versions of orthogonal sums. In order get a feeling for the differenceright now note that in the case of Hilb ( A ) for a unital C ∗ -algebra A as mentioned abovewe have End
Hilb ( A ) ( (cid:76) N A ) ∼ = B ( (cid:96) ⊗ A ) for the orthogonal sum according the our definition.In particular, (cid:76) N A admits an identity endomorphism. The AV-sum (cid:76) AV N A exists in theideal K ( A ) of compact operators in Hilb ( A ), and we have End K ( A ) ( (cid:76) AV N A ) ∼ = K ( (cid:96) ⊗ A ).The AV-sum (cid:76) AV N A is a non-unital object of K ( A ).Since C ∗ -categories are enriched over abelian groups it is clear that unital functors betweenunital C ∗ -categories preserve finite sums. The case of infinite sums is considerably morecomplicated. For example, if D is a unital closed subcategory of a C ∗ -category C , then ingeneral the orthogonal sum of a family of objects in D may not be isomorphic to the sumof the same family considered in C . Example 6.4 provides an example of this situation.In Section 6 we investigate the relation between these two sums in detail and provideconditions ensuring that they are unitarily isomorphic, see e.g. Proposition 6.3.In Section 7 we investigate the subcategories of C ∗ Cat of C ∗ -categories admitting sums ofa given cardinality and functors preserving such sums. We show that these subcategoriesare preserved by two-categorical limits and filtered colimits.The closure properties obtained in Section 7 may suggest that there are left-adjoint additivecompletion functors. But in Section 8 we show in Proposition 8.1, that in contrast to thecase of finite sums, infinite sum completion functors can not serve as left adjoints of aBousfield localization.Given a family of functors with target in a C ∗ -category we can form the orthogonal sumof these functors objectwise provided the target category admits the corresponding sums.This and related material is discussed in Section 9. In particular we use this sum of functorsin order to introduce the notion of flasqueness in Definition 9.3. In the equivariant case,since sums are only unique up to unique unitary isomorphism, forming sums of equivariantfunctors in general only yields a functor which is equivariant in a two-categorical sense.This will be captured in the notion of a weakly invariant functor, see Definition 9.6.Given a C ∗ -category ˜ C with an action of a group G in [Bun] we introduced the maximalcrossed product ˜ C (cid:111) G as the completion of an algebraic crossed product with respect to the This preprint appeared while we were finishing a first version of the present paper. C ∗ -categoryof homotopy G -orbits in C . As in the case of C ∗ -algebras besides the maximal one thereare also other choices for the completion. In general these choices are less functorial butanalytically more interesting. One such natural choice is the reduced crossed product˜ C (cid:111) r G which we define in Definition 10.8 and discuss in Section 10. As explained above,its construction heavily relies on our notion of infinite orthogonal sums in C ∗ -categories.Our reason for considering the reduced crossed product is twofold. First of all it appearsnaturally in the calculation of the values on discrete bornological coarse spaces of thecoarse homology theories constructed in [BE]. On the other hand, the functors on theorbit category which provide the topological side of the Baum–Connes assembly map (seeDefinition 17.19) involve the reduced crossed product in their construction. We will furtherextend the result that for amenable groups the maximal and reduced crossed products areisomorphic from C ∗ -algebras to C ∗ -categories.In Definition 11.5 of a homological functor we axiomatize some of the properties of the K -theory functor K C ∗ Cat for C ∗ -categories. The construction of coarse homology theoriesin [BE] only relies on these axioms. In Section 11 we further derive some immediateconsequences of the axioms like additivity or annihilation of flasques.In Section 12 we verify that the K -theory functor for C ∗ -categories K C ∗ Cat introduced by[Joa03] is indeed an example of a homological functor.In Theorem 13.7 we show that the K -theory functor for C ∗ -categories K C ∗ Cat preservesarbitrary products of additive C ∗ -category. This a special property of K -theory which wedo not expect for arbitrary homological functors. It is similar in spirit with the resultsshown in [Car95, KW17, KW19]. The fact that K C ∗ Cat preserves products is one of themain inputs for the proof of Theorem 17.29 provided in [BE].In Section 14 we consider the algebraic notion of Morita equivalences between C ∗ -categoriesintroduced in [DT14] and homological functors preserving them. In Theorem 14.14 weshow that the K -theory functor for C ∗ -categories K C ∗ Cat preserves Morita equivalences.Furthermore, in Proposition 14.8 we show that the reduced crossed product functorpreserves Morita equivalences.So far unitary equivalences, idempotent completion and Morita equivalences were consideredfor unital C ∗ -categories. In Section 15 we generalizes these notions to the relative situationof an ideal in an unital C ∗ -category. We further extend the corresponding rigidity propertiesof homological functors from the unital to the relative case.It is well-known that the left upper corner embedding of a unital C ∗ -algebra into thecompact operators on a free Hilbert module induces an equivalence in the K -theory of C ∗ -algebras. In Section 16 we generalize this situation by introducing the notion of aweak Morita equivalence between C ∗ -categories. As in the case of C ∗ -algebras it is acondition about the approximability of morphisms in the bigger category by conjugates ofmorphisms in the smaller. In particular, the notion of a weak Morita equivalence belongs5o the functional analytic corner of the field and has no counterpart in algebra. Our mainresult is Theorem 16.4 saying that the K-theory of C ∗ -categories K C ∗ Cat sends weak Moritaequivalences to equivalences. This result will be used in [BEL].The final Section 17 is devoted to the construction of functors on the orbit category startingfrom the data of a C ∗ -category with G -action and an invariant ideal on the one hand, andsome auxiliary functor Hg (e.g. K C ∗ Cat ) on the other. Using homotopy theoretic methodsfollowing [Bun19] we construct a functor Hg G ˜ K u , max whose values on orbits G/H are givenby Hg( ˜ K u (cid:111) H ) and involve the maximal crossed product, where ˜ K u is the G -subcategoryof unital objects in K . Using the theory of orthogonal sums and reduced crossed productsof C ∗ -categories with groups we provide an explicit construction of a functor Hg G ˜ K u ,r together with a comparison map Hg G ˜ K u , max → Hg G ˜ K u ,r which on orbits reduces to thecanonical morphism Hg( ˜ K u (cid:111) H ) → Hg( ˜ K u (cid:111) r H ) from the maximal to the reducedcrossed product. While the homotopy theoretic approach provides insights in the formalproperties of Hg G ˜ K u , max the functor Hg G ˜ K u ,r is relevant for the Baum–Connes assembly mapas discussed in [BEL] and the subject of one of the main results of [BE] reproduced hereas Theorem 17.29. We furthermore refer to Corollary 17.27 for an interesting applicationof the comparison map. Acknowledgement: U.B. was supported by the SFB 1085 (Higher Invariants) funded by theDeutsche Forschungsgemeinschaft (DFG). C -linear ∗ -categories and C ∗ -categories We start with the definitions of C -linear ∗ -categories and C ∗ -categories and their possiblynon-unital versions. We refer to [Bun19, Bun] for further details and the comparison withthe classical definitions [Bun19, Rem. 2.15].In order to fix set-theoretic size issues we fix a sequence of three Grothendieck universeswhose elements are called very small, small and large sets, respectively.A possibly non-unital small category C consists of a small set of objects Ob( C ), forevery two objects C, C (cid:48) a small set of morphisms
Hom C ( C, C (cid:48) ), and an associative law ofcomposition. A functor φ : C → D between two possibly non-unital categories is givenby a map between the sets of objects Ob( C ) → Ob( D ), and for every two objects C, C (cid:48) in C a map of morphism sets Hom C ( C, C (cid:48) ) → Hom D ( φ ( C ) , φ ( C (cid:48) )) which respects the lawsof compositions. The possibly non-unital small categories and functors form the largecategory of possibly non-unital small categories.A small category is a possibly non-unital small category which admits units for all itsobjects. A unital functor between categories is a functor which preserves units. We getthe large category of small categories and unital functors. It is a subcategory of the large6ategory of possibly non-unital small categories. The inclusion is neither full nor wide.A possibly non-unital small C -linear category is a possibly non-unital small category whichis enriched in C -vector spaces. Thus its morphism sets have the additional structure of C -vector spaces, and the composition laws are required to be bi-linear. Functors betweenpossibly non-unital small C -linear categories are required to respect the enrichment in C -vector spaces.A possibly non-unital small C -linear ∗ -category is a possibly non-unital small C -linearcategory equipped with an involution ∗ (a contravariant endofunctor of the underlyingpossibly non-unital category) fixing objects, reversing the direction of morphisms, andacting complex anti-linearly on the morphism spaces. A functor between possibly non-unital small C -linear ∗ -categories is a functor between possibly non-unital small C -linearcategories which in addition preserves the involutions. Definition 2.1.
We let ∗ Cat nu C denote the large category of possibly non-unital small C -linear ∗ -categories, and we let ∗ Cat C denote the subcategory of unital small C -linear ∗ -categories and unital functors. Example 2.2.
A non-unital ∗ -algebra over C can be considered as an object of ∗ Cat nu C which has a single object. An example is the ∗ -algebra of finite-rank operators on an ∞ -dimensional Hilbert space.The unital ∗ -algebras C and Mat (2 , C ) are objects of ∗ Cat C . The upper left corner inclusion C → Mat (2 , C ) is a morphism in ∗ Cat nu C , but not in ∗ Cat C .The category of very small Hilbert spaces and finite-rank linear operators Hilb fin-rk ( C )is an object of ∗ Cat nu C . The ∗ -operation is given by taking adjoints. Its full subcategory Hilb fg ( C ) of finite-dimensional Hilbert spaces is an object of ∗ Cat C .If H is a Hilbert space, then by B ( H ) we denote the C ∗ -algebra of bounded operators.It has a norm (cid:107) − (cid:107) B ( H ) . If H is very small, then we will consider B ( H ) as an object of ∗ Cat nu C . If C is in ∗ Cat nu C and f is a morphism in C , then we define (cid:107) f (cid:107) max := sup ρ (cid:107) ρ ( f ) (cid:107) B ( H ) , (2.1)where ρ runs over all functors ρ : C → B ( H ) for all very small complex Hilbert spaces H .Since there is at least the zero functor we know that (cid:107) f (cid:107) max takes values in [0 , ∞ ].Let C be in ∗ Cat nu C . Definition 2.3. C is called a pre- C ∗ -category if (cid:107) f (cid:107) max < ∞ for all morphisms f in C .We denote by ∗ pre Cat nu C and ∗ pre Cat C the full subcategories of ∗ Cat nu C and ∗ Cat C of pre- C ∗ -categories, respectively. C be in ∗ pre Cat nu C . Then (cid:107) − (cid:107) max induces semi-norms on the morphism spaces of C .A semi-normed complex vector space is said to be complete if the semi-norm is a norm,and the vector space is complete with respect to the metric induced by the norm. In thefollowing, completeness always refers to (cid:107) − (cid:107) max . Definition 2.4. C is called a C ∗ -category if the morphism spaces of C are complete.We denote by C ∗ Cat nu and C ∗ Cat the full subcategories of ∗ pre Cat nu C and ∗ pre Cat C of C ∗ -categories, respectively. Remark 2.5.
Classically the notions of a pre- C ∗ -algebra and a pre- C ∗ -category have adifferent meaning. A pre- C ∗ -algebra (in the classical sense) is a (sub-multiplicatively)normed *-algebra A such that the C ∗ -identity (cid:107) a ∗ a (cid:107) = (cid:107) a (cid:107) holds for all elements a of A .Then A is a C ∗ -algebra if it is in addition complete. Any pre- C ∗ -algebra A (in the classicalsense) can be completed to a C ∗ -algebra ¯ A . A selfadjoint element a in a pre- C ∗ -algebra iscalled positive if its image in ¯ A is positive, i.e., has a spectrum contained in [0 , ∞ ).Similarly, a (sub-multiplicatively) normed *-category C is a pre- C ∗ -category in the classicalsense [Mit02, Defn. 2.4] if the following conditions hold:1. The C ∗ -identity (cid:107) x (cid:107) = (cid:107) x ∗ x (cid:107) is satisfied for all morphisms x in Hom C ( A, B ) and forall objects
A, B of C .2. For every morphism x in Hom C ( A, B ) the morphism x ∗ x is a positive element of thepre- C ∗ -algebra Hom C ( A, A ).A C ∗ -category in the classical sense is then a pre- C ∗ -category in the classical sense whosemorphisms spaces are complete. Equivalently, by [Mit02, Thm. 2.7 & Defn. 2.9] one canrequire the following conditions:3. The C ∗ -inequality (cid:107) x (cid:107) ≤ (cid:107) x ∗ x + y ∗ y (cid:107) is satisfied for all morphisms x, y in Hom C ( A, B )and for all objects
A, B of C .4. For every morphism x in Hom C ( A, B ) the morphism x ∗ x is a positive element of thepre- C ∗ -algebra Hom C ( A, A ).In [Mit02, Ex. 2.10] Mitchener provides an example of a normed ∗ -category which satisfiesboth the C ∗ -identity and C ∗ -inequality, but not the positivity condition. Definition 2.6.
A normed *-category C satisfies the strong C ∗ -inequality if for all objects A, B, C of C and all morphisms x in Hom C ( A, B ) and y in Hom C ( A, C ) we have (cid:107) x (cid:107) ≤ (cid:107) x ∗ x + y ∗ y (cid:107) . (2.2)Note that the difference to Condition 3 above is that x and y may have different targets.8he strong C ∗ -inequality implies both the C ∗ -inequality and the C ∗ -identity, and it impliesthe positivity axiom by exploiting the following property of C ∗ -algebras: a self-adjointelement b in a C ∗ -algebra A is positive if and only if for all positive elements a in A wehave (cid:107) a (cid:107) ≤ (cid:107) a + b (cid:107) . On the other hand, this property of C ∗ -algebras also implies that thestrong C ∗ -inequality is true for the maximal norm on a pre- C ∗ -category in the sense ofDefinition 2.3.Since the norm on a C ∗ -category (in the classical sense) is equal to the maximal norm wesee that the definitions of a C ∗ -category in the sense of Definition 2.4 and in the classicalsense are equivalent. Example 2.7. A C ∗ -algebra is a C ∗ -category with a single object. It could be unital ornon-unital. If, according to the classical definition, a C ∗ -algebra is considered as a closed ∗ -subalgebra A of B ( H ) for some Hilbert space H , then the maximal norm on A coincideswith the restriction of the usual operator norm from B ( H ) to A .If A is a C ∗ -algebra, then the category of very small A -Hilbert- C ∗ -modules Hilb ( A ) andbounded adjointable operators is an object of C ∗ Cat . It contains the wide subcategory K ( A ) whose morphisms are compact operators (in the sense of A -Hilbert C ∗ -modules).The inclusion K ( A ) → Hilb ( A ) is a functor in C ∗ Cat nu . Example 2.8.
The functors from ∗ Cat nu C , ∗ Cat C , C ∗ Cat nu and C ∗ Cat to small setswhich take the sets of objects, have right-adjoints, see [Bun, Lem. 2.4 and 3.8]. In all casesthe right-adjoint 0[ − ] sends a set X to the category 0[ X ] with the set of objects X , andwhose morphism vector spaces are all trivial. We have a canonical counit functor C → C )] (2.3)in all of these cases.Let C be in C ∗ Cat and f be a morphism in C . Recall the definition of the maximal norm(2.1). It is an immediate consequence of the definition of the maximal norm that (cid:107) σ ( f ) (cid:107) max ≤ (cid:107) f (cid:107) max (2.4)for every morphism σ : C → C (cid:48) in C ∗ Cat nu . In the following we show that the maximalnorm of a morphism in a unital C ∗ -category can be generated by unital representations inthe object Hilb ( C ) of C ∗ Cat of very small Hilbert spaces and bounded linear operators.We use the notation (cid:107) − (cid:107) in order to denote the operator norm of bounded operatorsbetween Hilbert spaces. Of course it coincides with the maximal norm on
Hilb ( C )considered as a pre- C ∗ -category. This was already stated in [Bun19, Rem. 2.15], but one must delete the word “parallel” in the statementof the C ∗ -inequality in order to turn it into the strong C ∗ -inequality. emma 2.9. We have (cid:107) f (cid:107) max = sup σ ∈ Hom C ∗ Cat ( C , Hilb ( C )) (cid:107) σ ( f ) (cid:107) .Proof. From (2.4) we get that sup σ (cid:107) σ ( f ) (cid:107) ≤ (cid:107) f (cid:107) max . It therefore suffices to show that (cid:107) f (cid:107) max ≤ sup σ (cid:107) σ ( f ) (cid:107) . Let ρ : C → B ( H ρ ) be a functor in C ∗ Cat nu . Since C is unital,using the adjunction [Bun, (3.10)] we get a unital functor ˆ ρ : C → Hilb ( C ). It sendsan object C in C to the image ˆ ρ ( C ) := ρ ( id C )( H ρ ) of the projection ρ ( id C ), and amorphism f : C → C (cid:48) to the morphism ρ ( id C (cid:48) ) f | ˆ ρ ( C ) : ˆ ρ ( C ) → ˆ ρ ( C (cid:48) ). Note that in contrastto ρ the functor ˆ ρ is not constant on objects anymore. By an inspection we see that (cid:107) ρ ( f ) (cid:107) = (cid:107) ˆ ρ ( f ) (cid:107) . Consequently, (cid:107) f (cid:107) max = sup ρ (cid:107) ρ ( f ) (cid:107) = sup ρ (cid:107) ˆ ρ ( f ) (cid:107) ≤ sup σ (cid:107) σ ( f ) (cid:107) . Let C in ∗ Cat nu C and let C be an object of C . Definition 2.10. C is unital if there exists an identity morphism in End C ( C ) . By C u we denote the full subcategory of unital objects in C . Note that C u is an object of ∗ Cat C . If C is in C ∗ Cat nu , then C u is in C ∗ Cat .Unital objects are preserved by automorphisms. Therefore, if G is a group and ˜ C isin Fun ( BG, ∗ Cat nu C ) or in Fun ( BG, C ∗ Cat nu ), then we naturally get an object ˜ C u in Fun ( BG, ∗ Cat C ) or Fun ( BG, C ∗ Cat ), respectively.In the following we consider properties of morphisms in C . Definition 2.11.
1. A projection is an endomorphism p such that p ∗ = p and p = p .2. A partial isometry is a morphism u such that uu ∗ and u ∗ u are projections.3. An isometry is a partial isometry u : C → C (cid:48) such that u ∗ u = id C .4. A unitary is an isometry u : C → C (cid:48) such that uu ∗ = id C (cid:48) . Remark 2.12.
Note that the condition p ∗ = p in Definition 2.11(1) describes orthogonalprojections. In the present paper we will only consider orthogonal projections and thereforedrop the adjective “orthogonal”.If u : C → C (cid:48) is an isometry, then implicitly the object C is unital. Similarly, unitariescan only exist between unital objects. 10et p be a projection on C . Definition 2.13.
An image of p is a pair ( D, u ) of an object D in C and an isometry u : D → C such that p = uu ∗ . Lemma 2.14.
The image of a projection is uniquely determined up to unique unitaryisomorphism.Proof.
Let (
D, u ) and ( D (cid:48) , u (cid:48) ) be both images of p . Then v := u (cid:48) , ∗ u : D → D (cid:48) is the uniqueunitary such that u (cid:48) v = u . Definition 2.15.
1. A projection is called effective if it admits an image.2. C is called idempotent complete if every projection in C is effective. Example 2.16. If A is a C ∗ -algebra, then Hilb ( A ) is idempotent complete.The full subcategory Hilb dim= ∞ ( C ) of ∞ -dimensional Hilbert spaces in Hilb ( C ) is anexample which is not idempotent complete. C ∗ -categories The notion of a finite orthogonal sum of objects in a unital C ∗ -category can be definedin the standard way. In the present section we are mainly interested in infinite sums ofobjects in C ∗ -categories. After briefly recalling the finite case [DT14] we introduce ournotion of an orthogonal sum of an arbitrary family of objects in a unital C ∗ -category.In the case of the category of Hilbert C ∗ -modules over a C ∗ -algebra we check that ourdefinition of an orthogonal sum is equivalent to the classical definition.Let C be in ∗ Cat nu C , and let ( e i ) i ∈ I be a family of morphisms C i → C in C with the sametarget. Definition 3.1.
The family ( e i ) i ∈ I is mutually orthogonal if for all i, j in I with i (cid:54) = j wehave e ∗ j e i = 0 . Let ( C i ) i ∈ I be a finite family of objects in C .11 efinition 3.2. An orthogonal sum of the family ( C i ) i ∈ I is a pair ( C, ( e i ) i ∈ I ) of an object C in C and a family of isometries e i : C i → C such that:1. The family ( e i ) i ∈ I is mutually orthogonal.2. (cid:80) i ∈ I e i e ∗ i = id C . Note that by this definition only families of unital objects can admit orthogonal sums.The sum of such a family is also unital.
Example 3.3.
The orthogonal sum of an empty family is a zero object.If ( C, ( e i ) i ∈ I ) is an orthogonal sum of the finite family ( C i ) i ∈ I , then it represents thecategorical coproduct of the family ( C i ) i ∈ I . The pair ( C, ( e ∗ i ) i ∈ I ) represents the categorialproduct of the family ( C i ) i ∈ I . In particular, an orthogonal sum is uniquely determined upto unique isomorphism. The following lemma qualifies this isomorphism to be unitary. Lemma 3.4.
An orthogonal sum of a finite family is unique up to unique unitary isomor-phism.Proof.
Let ( C, ( e i ) i ∈ I ) and ( C (cid:48) , ( e (cid:48) i ) i ∈ I ) be two orthogonal sums of the finite family ( C i ) i ∈ I .Then v := (cid:80) i ∈ I e (cid:48) i e ∗ i : C → C (cid:48) is the unique isomorphism such that ve j = e (cid:48) j for all j in I .One easily checks by a calculation that it is unitary.Let C be in ∗ Cat nu C . Definition 3.5. C is additive if it admits orthogonal sums for all finite families. If C is additive, then it is unital since it must admit sums of all one-member families. Example 3.6. If A is a C ∗ -algebra, then the full subcategory Hilb fg ( A ) of Hilb ( A ) offinitely generated A -Hilbert- C ∗ -modules is additive.For the discussion of infinite orthogonal sums we specialize to unital C ∗ -categories. Let C be in C ∗ Cat .Note that a C ∗ -category is enriched in Banach spaces. We let Ban denote the categoryof complex Banach spaces and continuous linear maps. An object C in C (co)representsfunctors Hom C ( − , C ) : C op → Ban , Hom C ( C, − ) : C → Ban . C, ( e i ) i ∈ I ) be a pair consisting of an object C of C and a mutually orthogonal familyof isometries e i : C i → C . Using this data we are going to define two subfunctors K ( − , C ) : C op → Ban , K ( C, − ) : C → Ban of Hom C ( − , C ), or of Hom C ( C, − ), respectively.Let D be an object of C . A morphism f : D → C of the form f = e i ˜ f for some morphism˜ f : D → C i is called a generator for K ( D, C ). Similarly, a morphism f (cid:48) : C → D of theform f (cid:48) = f (cid:48) i e ∗ i for some morphism f (cid:48) i : C i → D is a called a generator for K ( C, D ).A finite linear combination of generators will be called finite. One checks that the subspacesof finite morphisms form subfunctors
Hom fin C ( − , C ) and Hom fin C ( C, − ) of Hom C ( − , C ) and Hom C ( C, − ), respectively, considered as C -vector space valued functors. Definition 3.7.
We define the subfunctors K ( − , C ) : C op → Ban , K ( C, − ) : C → Ban of Hom C ( − , C ) and Hom C ( C, − ) by taking objectwise the norm closures of Hom fin C ( − , C ) and Hom fin C ( C, − ) . In order to see that these subfunctors are well-defined we use the sub-multiplicativity ofthe norm on C in order to check these subspaces are preserved by precompositions orpostcompositions with morphisms in C , respectively. The involution of C provides anantilinear isomorphism between K ( C, D ) and K ( D, C ) for all D in C .Note that K ( C, D ) and K ( D, C ) depend on C and the family ( e i ) i ∈ I . If we want to stressthe dependence of these subspaces on the family ( e i ) i ∈ I , then we will write K (( C, ( e i ) i ∈ I ) , D )and K ( D, ( C, ( e i ) i ∈ I )). This notation will in particular be used in order to avoid confusionif we want to consider the case where D = C . Example 3.8.
For i in I the morphism e i : C i → C belongs to K ( C i , C ). It is actually agenerator. Similarly, e ∗ i is a generator in K ( C, C i ). Example 3.9.
Let A be a C ∗ -algebra and consider the C ∗ -category Hilb ( A ) of Hilbert A -modules and continuous adjointable operators. Let C be in Hilb ( A ) and assume that( e i ) i ∈ I is a mutually orthogonal family of isometries e i : C i → C . Furthermore, assume thatthe images of the morphisms e i together generate C as an A -Hilbert C ∗ -module. Then wehave inclusions K ( D, C ) ⊆ K ( D, C ) , K ( C, D ) ⊆ K ( C, D ) , (3.1)where K ( D, C ) and K ( C, D ) denote the spaces of all compact operators (in the sense of A -Hilbert C ∗ -modules) from D to C and vice versa. In general, these inclusions are proper.But if id C i is compact for every i in I , then the inclusions in (3.1) are equalities.13et C be a category, let F, F (cid:48) : C →
Ban be two functors, and let u : F → F (cid:48) be a naturaltransformation. The latter is given by a family u = ( u C ) C ∈ Ob( C ) of continuous maps u C : F ( C ) → F (cid:48) ( C ). Definition 3.10.
The transformation u is called uniformly bounded if (cid:107) u (cid:107) := sup C ∈C (cid:107) u C (cid:107) Hom
Ban ( F ( C ) ,F (cid:48) ( C )) < ∞ . We let
Hom bd Fun ( C , Ban ) ( F, F (cid:48) ) denote the subspace of
Hom
Fun ( C , Ban ) ( F, F (cid:48) ) of all uniformlybounded natural transformations. One easily checks that
Hom bd Fun ( C , Ban ) ( F, F (cid:48) ) is a Banachspace with norm (cid:107) − (cid:107) .We let ( C, ( e i ) i ∈ I ) be as above and fix an object D . Definition 3.11.
1. The Banach space of right multipliers from D to C is defined by RM(
D, C ) :=
Hom bd Fun ( C , Ban ) ( K ( C, − ) , Hom C ( D, − )) .
2. The Banach space of left multipliers from C to D is defined by LM(
C, D ) :=
Hom bd Fun ( C op , Ban ) ( K ( − , C ) , Hom C ( − , D )) . If we want to stress the dependence of these objects on the family ( e i ) i ∈ I or insert C inplace of D , then we will write RM( D, ( C, ( e i ) i ∈ I )) and LM(( C, ( e i ) i ∈ I ) , D ).A right multiplier in RM( D, C ) is given by a uniformly bounded family R = ( R D (cid:48) ) D (cid:48) ∈ Ob( C ) of bounded linear maps R D (cid:48) : K ( C, D (cid:48) ) → Hom C ( D, D (cid:48) ) satisfying the conditions for a natu-ral transformation. In particular, for a morphism f in Hom C ( D (cid:48) , D (cid:48)(cid:48) ) and g in K ( C, D (cid:48) ) wehave R D (cid:48)(cid:48) ( f g ) = f R D (cid:48) ( g ). We use a similar notation ( L D (cid:48) ) D (cid:48) ∈ Ob( C ) for left multipliers. Theinvolution of C induces an antilinear isomorphism between RM( D, C ) and LM(
C, D ).Let ( C i ) i ∈ I be a family of objects of C , and let ( h i ) i ∈ I and ( h (cid:48) i ) i ∈ I be families of morphisms h i : D → C i and h (cid:48) i : C i → D in C . Definition 3.12.
1. We say that ( h i ) i ∈ I is square summable if sup J ⊆ I (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i h i (cid:13)(cid:13) < ∞ , (3.2) where J runs over the set of finite subsets of I . . We say that ( h (cid:48) i ) i ∈ I is square summable if sup J ⊆ I (cid:13)(cid:13) (cid:88) i ∈ J h (cid:48) i h (cid:48) , ∗ i (cid:13)(cid:13) < ∞ , (3.3) where J runs over the set of finite subsets of I . Lemma 3.13.
1. If the family ( h ∗ i ) i ∈ I is mutually orthogonal and uniformly bounded, then ( h i ) i ∈ I issquare summable.2. If the family ( h (cid:48) i ) i ∈ I is mutually orthogonal and uniformly bounded, then it is squaresummable.Proof. We show Assertion 1. Assertion 2 can then be deduced using the involution.Let J be a finite subset of I . Using the fact that ( h ∗ i ) i ∈ I is mutually orthogonal we calculatethat for every k in N (cid:0) (cid:88) i ∈ J h ∗ i h i (cid:1) k = (cid:88) i ∈ J ( h ∗ i h i ) k . Using repeatedly the C ∗ -equality, that (cid:80) i ∈ J h ∗ i h i is self-adjoint, and the triangle inequalityfor the norm we get (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i h i (cid:13)(cid:13) k = (cid:13)(cid:13)(cid:0) (cid:88) i ∈ J h ∗ i h i (cid:1) k (cid:13)(cid:13) = (cid:13)(cid:13) (cid:88) i ∈ J ( h ∗ i h i ) k (cid:13)(cid:13) ≤ (cid:88) i ∈ J (cid:107) h ∗ i h i (cid:107) k ≤ | J | max i ∈ J (cid:107) h ∗ i h i (cid:107) k . We take the 2 k -th root and form the limit for k → ∞ . Using that lim k →∞ | J | k = 1 weget (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i h i (cid:13)(cid:13) ≤ max i ∈ J (cid:107) h ∗ i h i (cid:107) ≤ sup i ∈ I (cid:107) h i (cid:107) . Since the right-hand side is finite by assumption and does not depend on J , we concludethe square summability of h .The following lemma provides a tool to construct interesting multipliers. Let ( C, ( e i ) i ∈ I )be a pair consisting of an object C of C and a mutually orthogonal family of isometries e i : C i → C . Let h := ( h i ) i ∈ I be a family of morphisms h i : D → C i , and let h (cid:48) := ( h (cid:48) i ) i ∈ I be a family of morphisms h (cid:48) i : C i → D . Lemma 3.14. . There exists a right-multiplier R ( h ) in RM(
D, C ) with R ( h ) C i ( e ∗ i ) = h i for all i in I if and only if h is square summable.If h is square summable, then R ( h ) is uniquely determined and its norm is given by (cid:107) R ( h ) (cid:107) = (cid:115) sup J ⊆ I (cid:107) (cid:88) i ∈ J h ∗ i h i (cid:107) . (3.4)
2. There exists a left-multiplier L ( h (cid:48) ) in LM(
C, D ) with L ( h (cid:48) ) C i ( e i ) = h (cid:48) i for all i in I if and only if h (cid:48) is square summable.If h (cid:48) is square summable, then L ( h (cid:48) ) is uniquely determined and its norm is given by (cid:107) L ( h (cid:48) ) (cid:107) = (cid:115) sup J ⊆ I (cid:107) (cid:88) i ∈ J h (cid:48) i h (cid:48) , ∗ i (cid:107) . (3.5) Proof.
It suffices to prove Assertion 1. Assertion 2 then follows from 1 using the involution.We first assume that h is square summable. Let J be a finite subset of I and consider R J ( h ) := (cid:88) j ∈ J e j h j (3.6)in Hom C ( D, C ). Right composition with R J ( h ) provides a right-multiplier ( R J ( h ) D (cid:48) ) D (cid:48) ∈ Ob( C ) such that R J ( h ) D (cid:48) sends f in K ( C, D (cid:48) ) to (cid:80) j ∈ J f e j h j in Hom C ( D, D (cid:48) ). In particular, wehave R J ( h ) C i ( e ∗ i ) = h i provided i ∈ J .We now show that lim J ⊆ I R J ( h ) exists pointwise on finite morphisms and defines therequired multiplier R ( h ) by continuous extension. Here the limit is taken over the filteredposet of finite subsets of I . So let D (cid:48) be in Ob( C ) and let f be in Hom fin C ( C, D (cid:48) ). Then J (cid:55)→ R J ( h ) D (cid:48) ( f ) is eventually constant and the limit lim J ⊆ I R J ( h ) D (cid:48) ( f ) exists. We get anatural transformation lim J ⊆ I R J ( h ) : Hom fin C ( C, − ) → Hom C ( D, − )of C -vector space valued functors. We now argue that lim J ⊆ I R J ( h ) extends by continuityto a natural transformation R ( h ) : K ( C, − ) → Hom C ( D, − ) . To this end it suffices to verify that lim J ⊆ I R J ( h ) is bounded for any D (cid:48) in Ob( C ) separately.Let f be in Hom fin ( C, D (cid:48) ). Then for sufficiently large finite subsets J f of I we calculateusing the sub-multiplicativity of the norm under composition, the C ∗ -equality for the norm16n a C ∗ -category, the mutual orthogonality of the family ( e i ) i ∈ I , and that e ∗ i e i = id C i forevery i in I that (cid:107) lim J ⊆ I R J ( h ) D (cid:48) ( f ) (cid:107) = (cid:13)(cid:13) f (cid:88) j ∈ J f e j h j (cid:13)(cid:13) ≤ (cid:107) f (cid:107) · (cid:13)(cid:13) (cid:88) j ∈ J f e j h j (cid:13)(cid:13) (3.7)= (cid:107) f (cid:107) · (cid:13)(cid:13)(cid:0) (cid:88) i ∈ J f e i h i (cid:1) ∗ (cid:0) (cid:88) j ∈ J f e j h j (cid:1)(cid:13)(cid:13) = (cid:107) f (cid:107) · (cid:13)(cid:13) (cid:88) i ∈ J f h ∗ i e ∗ i e i h i (cid:13)(cid:13) = (cid:107) f (cid:107) · (cid:13)(cid:13) (cid:88) i ∈ J f h ∗ i h i (cid:13)(cid:13) ≤ (cid:107) f (cid:107) · sup J ⊆ I (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i h i (cid:13)(cid:13) , where the supremum runs over all finite subsets of I . By Assumption (3.2) it follows that lim J ⊆ I R J ( h ) D (cid:48) is bounded. Since the right-hand side does not depend on D (cid:48) we furthersee that lim J ⊆ I R J ( h ) is uniformly bounded. The above estimate also implies that (cid:107) R ( h ) (cid:107) ≤ sup J ⊆ I (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i h i (cid:13)(cid:13) . (3.8)The converse estimate implying the equality (3.4) will be shown below while proving theconverse to the existence statement.We now assume the existence of a right-multiplier R ( h ) in RM( D, C ) with R ( h ) C i ( e ∗ i ) = h i for all i in I and verify that h is square summable. So let J be a finite subset of I . Then R ( h ) D (cid:0) (cid:88) i ∈ J h ∗ i e ∗ i (cid:1) = (cid:88) i ∈ J h ∗ i R ( h ) C i ( e ∗ i ) = (cid:88) i ∈ J h ∗ i h i and consequently (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i h i (cid:13)(cid:13) = (cid:13)(cid:13) R ( h ) D (cid:0) (cid:88) i ∈ J h ∗ i e ∗ i (cid:1)(cid:13)(cid:13) ≤ (cid:107) R ( h ) (cid:107) · (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i e ∗ i (cid:13)(cid:13) . Using the involution we get a left-multiplier R ( h ) ∗ in LM( C, D ) with R ( h ) ∗ C i ( e i ) = h ∗ i forall i in I . It satisfies (cid:88) i ∈ J h ∗ i e ∗ i = (cid:88) i ∈ J R ( h ) ∗ C i ( e i ) e ∗ i = (cid:88) i ∈ J R ( h ) ∗ C ( e i e ∗ i ) = R ( h ) ∗ C (cid:0) (cid:88) i ∈ J e i e ∗ i (cid:1) and consequently (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i e ∗ i (cid:13)(cid:13) = (cid:13)(cid:13) R ( h ) ∗ C (cid:0) (cid:88) i ∈ J e i e ∗ i (cid:1)(cid:13)(cid:13) ≤ (cid:107) R ( h ) ∗ (cid:107) · (cid:13)(cid:13) (cid:88) i ∈ J e i e ∗ i (cid:13)(cid:13) ≤ (cid:107) R ( h ) ∗ (cid:107) , where the last inequality sign holds because ( e i e ∗ i ) i ∈ I is a mutually orthogonal family ofprojections. Putting all together, we conclude the inequality (cid:13)(cid:13) (cid:88) i ∈ J h ∗ i h i (cid:13)(cid:13) ≤ (cid:107) R ( h ) (cid:107) for every finite subset J of I . This implies square summability of h . Furthermore, since h ∗ i h i is a positive operator for every i in I it also implies the converse inequality to (3.8)which finishes the verification of the equality (3.4).17he following maps will play a crucial role in the characterization of infinite orthogonalsums. They send morphisms to the corresponding multipliers. Definition 3.15.
For every object D in C we define the associated right multiplier map m RD : Hom C ( D, C ) → Hom bd Fun ( C , Ban ) ( Hom C ( C, − ) , Hom C ( D, − )) (3.9) → Hom bd Fun ( C , Ban ) ( K ( C, − ) , Hom C ( D, − )) = RM( D, C ) . Similarly we define the associated left multiplier map m LD : Hom C ( C, D ) → Hom bd Fun ( C op , Ban ) ( Hom C ( − , C ) , Hom C ( − , D )) (3.10) → Hom bd Fun ( C op , Ban ) ( K ( − , C ) , Hom C ( − , D )) = LM( C, D ) . We have m R := ( m RD ) D ∈ Ob( C ) ∈ Hom bd Fun ( C op , Ban ) ( Hom C ( − , C ) , RM( − , C )) , (cid:107) m R (cid:107) ≤ m L := ( m LD ) D ∈ Ob( C ) ∈ Hom bd Fun ( C , Ban ) ( Hom C ( C, − ) , LM( C, − )) , (cid:107) m L (cid:107) ≤ , (3.12)where the norm estimate follows from the sub-multiplicativity of the norm on C .We can now define the notion of an orthogonal sum of a family ( C i ) i ∈ I of objects in C . Definition 3.16.
An orthogonal sum of the family ( C i ) i ∈ I is a pair ( C, ( e i ) i ∈ I ) of an object C in C together with a mutually orthogonal family of isometries e i : C i → C such that theassociated multiplier transformations (3.9) and (3.10) are bijective for any object D of C . Remark 3.17.
Note that the associated multiplier transformations (3.9) and (3.10) arecontinuous linear maps between Banach spaces. Therefore, if they are bijective thenby the open mapping theorem their inverses are also continuous. In Proposition 3.22below we will show that bijectivity implies isometry. In this case it then follows that thefamilies of inverses ( m R, − D ) D ∈ Ob( C ) and ( m L, − D ) D ∈ Ob( C ) are uniformly bounded, and thetransformations m R and m L in (3.11) and (3.12) are isomorphisms as well. Remark 3.18. If I is a finite set, then it is an easy exercise to show that the notion ofan orthogonal sum according to Definition 3.16 coincides with the notion of an orthogonalsum according to Definition 3.2. Lemma 3.19.
An orthogonal sum of a family of objects is unique up to unique unitaryisomorphism. roof. Let ( C, ( e i ) i ∈ I ) and ( C (cid:48) , ( e (cid:48) i )) i ∈ I be two orthogonal sums of the family ( C i ) i ∈ I . Inthe following argument we will use repeatedly that the associated multiplier maps arebijective.By the Lemmas 3.13 and 3.14 the family e (cid:48) := ( e (cid:48) i ) i ∈ I induces a left multiplier L ( e (cid:48) ) inLM(( C, ( e i ) i ∈ I ) , C (cid:48) ) satisfying L ( e (cid:48) )( e i ) = e (cid:48) i for all i in I . It is the associated left multiplierof a uniquely determined morphism v : C → C (cid:48) satisfying ve i = e (cid:48) i for all i in I .Analogously, the family e := ( e i ) i ∈ I defines a left multiplier L ( e ) in LM(( C (cid:48) , ( e (cid:48) i ) i ∈ I ) , C )satisfying L ( e )( e (cid:48) i ) = e i for all i in I which is the associated left mulitplier of a uniquelydetermined morphism w : C (cid:48) → C such that we (cid:48) i = e i for all i in I .The associated left multiplier of vw is v ∗ L ( e ) in LM(( C (cid:48) , ( e (cid:48) i ) i ∈ I ) , C (cid:48) ) satisfying( v ∗ L ( e ))( e (cid:48) i ) = ve i = e (cid:48) i for all i in I . Since the associated left multiplier of id C (cid:48) has the same property we concludethat vw = id C (cid:48) . In a similar manner we show that wv = id C .We finally argue now that w = v ∗ . For every i in I and f in Hom fin C ( C, D ) we have for asufficiently large finite subset J of I , using f = (cid:80) j ∈ J f e j e ∗ j , f v ∗ e (cid:48) i = ( e (cid:48) , ∗ i vf ∗ ) ∗ = ( e (cid:48) , ∗ i v (cid:88) j ∈ J e j e ∗ j f ∗ ) ∗ = ( e (cid:48) , ∗ i (cid:88) j ∈ J e (cid:48) j e ∗ j f ∗ ) ∗ = ( e ∗ i f ∗ ) ∗ = f e i = f we (cid:48) i . This implies v ∗ e (cid:48) i = we (cid:48) i for all i in I . By the injectivity of the associated left multipliermap (3.10) we get v ∗ = w .The following two lemmas prepare the proof of Proposition 3.22 which states that for anorthogonal sum the associated multiplier maps (3.9) and (3.10) are isometric.Let A be in C ∗ Alg and I be a left-ideal in A . Recall ([Mur90, Thm. 3.1.2]) that left-idealsadmit approximate right-units, i.e., there is a net ( u ν ) ν ∈ N of positive elements of I with lim ν xu ν = x for every x in I . For an element a of A we define its right-multiplier normon I by (cid:107) a (cid:107) R ( I ) := sup x ∈ I, (cid:107) x (cid:107)≤ (cid:107) xa (cid:107) . A family of elements ( v κ ) κ ∈ K of A is called right-essential if for every non-zero a in A exists some κ in K such that av κ is not zero. We define the notion of a left-essential subsetanalogously using multiplication from the left. Lemma 3.20.
If the approximate unit ( u ν ) ν ∈ N of I is right-essential in A , then for every a in A we have (cid:107) a (cid:107) = (cid:107) a (cid:107) R ( I ) . roof. The inequality (cid:107) a (cid:107) R ( I ) ≤ (cid:107) a (cid:107) immediately follows from the sub-multiplicativity ofthe norm on A . We now show the reverse inequality .We let A ∗∗ be the von Neumann algebra given by the double commutant of the image of A under its universal representation. The weak closure of I in A ∗∗ will be denoted by I ∗∗ .It is a weakly closed left-ideal in A ∗∗ and therefore of the form A ∗∗ π I for some projection π I in A ∗∗ . In fact π I is the strong limit of ( u ν ) ν ∈ N in A ∗∗ . We refer to [Bla06, III.1.1.13]for these statements.We let ¯ I denote the strong closure of I in A ∗∗ . By [Mur90, Thm. 4.2.7] we know that ¯ I isalso weakly closed. Hence the canonical inclusion ¯ I ⊆ I ∗∗ is an equality.Since we assume that ( u ν ) ν ∈ N is right-essential we can conclude that the map A → A ∗∗ , a (cid:55)→ aπ I is injective. Hence its extension to a map A ∗∗ → A ∗∗ , z (cid:55)→ zπ I , is also injective[Bla06, III.5.2.10]. But this implies that π I = 1 A ∗∗ and therefore ¯ I = I ∗∗ = A ∗∗ .For every a in A we have the chain of equalities (cid:107) a (cid:107) R ( I ) = sup x ∈ I, (cid:107) x (cid:107)≤ (cid:107) xa (cid:107) ! = sup y ∈ ¯ I, (cid:107) y (cid:107)≤ (cid:107) ya (cid:107) !! = sup y ∈ A ∗∗ , (cid:107) y (cid:107)≤ (cid:107) ya (cid:107) , where in the equality marked ! we use that ¯ I is the strong closure of I in A ∗∗ and that A → A ∗∗ is isometric, and the equality marked by !! follows from I ∗∗ = A ∗∗ as shownabove.Since A ∗∗ is a von Neumann algebra it admits a measurable function calculus for self-adjointoperators. For any ε in (0 , ∞ ) we can define the projection q := 1 [ (cid:107) a (cid:107)− ε, (cid:107) a (cid:107) ] ( | aa ∗ | / ) in A ∗∗ .Since sup σ ( | aa ∗ | / ) = (cid:107) a (cid:107) we have σ ( | a ∗ a | / ) ∩ [ (cid:107) a (cid:107) − ε, (cid:107) a (cid:107) ] (cid:54) = ∅ and therefore q (cid:54) = 0.The spectral theorem implies the inequality aa ∗ ≥ ( (cid:107) a (cid:107) − ε ) q of self-adjoint operators.By [Mur90, Thm. 2.2.5(2)]) we then also have the inequality qaa ∗ q ≥ ( (cid:107) a (cid:107) − ε ) q . Usingthe C ∗ -identity for the first equality we therefore get the estimate (cid:107) qa (cid:107) = (cid:107) qaa ∗ q (cid:107) ≥ ( (cid:107) a (cid:107) − ε ) (cid:107) q (cid:107) q (cid:54) =0 = ( (cid:107) a (cid:107) − ε ) . Finally we get (cid:107) a (cid:107) R ( I ) = sup y ∈ A ∗∗ , (cid:107) y (cid:107)≤ (cid:107) ya (cid:107) ≥ (cid:107) qa (cid:107) ≥ (cid:107) a (cid:107) − ε . Since ε was arbitrary, the desired inequalilty (cid:107) a (cid:107) R ( I ) ≥ (cid:107) a (cid:107) follows. Remark 3.21.
Since the members of the net ( u ν ) ν ∈ N are positive and therefore self-adjoint, the assumption of Lemma 3.20 is equivalent to the assumption that this net is The argument is a modification of the argument that Ozawa provided to answer the MathOverflowquestion [Oza20]. By [Bla06, III.5.2.7] it is also isometrically isomorphic to the double dual of A considered as a Banachspace. This explains the notation A ∗∗ . (cid:107) a (cid:107) = (cid:107) a (cid:107) L ( I ) , where (cid:107) a (cid:107) L ( I ) := sup x ∈ I, (cid:107) x (cid:107)≤ (cid:107) ax (cid:107) is the norm of a considered as a left-multiplier on I .Let C be in C ∗ Cat . Let ( C i ) i ∈ I be a family of objects in C and assume that ( C, ( e i ) i ∈ I )represents an orthogonal sum of ( C i ) i ∈ I according to Definition 3.16. This is equivalent tothe fact that the associated multiplier maps (3.9) and (3.10) are bijective for every object D in C . Proposition 3.22.
The associated multiplier maps (3.9) and (3.10) are isometric forevery objects D in C .Proof. We only discuss the case of the associated right muliplier map (3.9). Then the caseof left multipliers (3.10) can be deduced using the involution of C . Let h : D → C be amorphism, and denote by R ( h ) := m RD ( h ) the associated right multiplier. The estimate in(3.11) immediately implies (cid:107) R ( h ) (cid:107) ≤ (cid:107) h (cid:107) .In order to show the reverse estimate, we claim that it suffices to prove it for endomorphismsof the object C . To show the claim assume that (cid:107) f (cid:107) ≤ (cid:107) R ( f ) (cid:107) for all endomorphisms f of C . Applying this to f = hh ∗ with h (cid:54) = 0 (the case h = 0 is obvious) we conclude (cid:107) hh ∗ (cid:107) ≤ (cid:107) R ( hh ∗ ) (cid:107) . Using the C ∗ -identity and that the involution is an isometry, wefurther get the equality (cid:107) hh ∗ (cid:107) = (cid:107) h ∗ (cid:107) = (cid:107) h ∗ (cid:107)(cid:107) h (cid:107) . On the other hand, by the definitionof the right multiplier norm we have the inequality (cid:107) R ( hh ∗ ) (cid:107) ≤ (cid:107) R ( h ) (cid:107)(cid:107) h ∗ (cid:107) . Combiningeverything and dividing by (cid:107) h ∗ (cid:107) , we arrive at the desired inequality (cid:107) h (cid:107) ≤ (cid:107) R ( h ) (cid:107) .In order to show (cid:107) f (cid:107) ≤ (cid:107) R ( f ) (cid:107) for every endomorphism f of C we employ Lemma 3.20.Recall that K (( C, ( e i ) i ∈ I ) , C ) is generated by morphisms f (cid:48) : C → C of the form f (cid:48) = f (cid:48) i e ∗ i for some morphism f (cid:48) i : C i → C . It follows that we have the inclusion Hom C ( C, C ) · K (( C, ( e i ) i ∈ I ) , C ) ⊆ K (( C, ( e i ) i ∈ I ) , C ) , i.e., K (( C, ( e i ) i ∈ I ) , C ) is a left-ideal in the C ∗ -algebra Hom C ( C, C ). For every finite subset J of I we define p J := (cid:80) i ∈ J e i e ∗ i in End C ( C ). It is immediate that the family ( p J ) J with J running through the poset of finite subsets of I is an approximate right-unit for K (( C, ( e i ) i ∈ I ) , C ). In order to apply Lemma 3.20 we must check that ( p J ) J is right-essentialin End C ( C ). Indeed, if f be a non-zero morphism in in End C ( C ), then by the injectivityof the associated left multiplier map there is an i in I such that f e i , and hence also f p { i } ,is non-zero.Let A be a unital C ∗ -algebra. Let Hilb ( A ) be the C ∗ -category of Hilbert A -modules as inExample 2.7. 21 emma 3.23. The notion of an orthogonal sum in
Hilb ( A ) according to Definition 3.16coincides with the classical notion.Proof. Let ( C i ) i ∈ I be a family in Hilb ( A ). We first recall the classical construction of theorthogonal sum of this family. We start with choosing an algebraic direct sum C alg := (cid:77) i ∈ C C i of A -right-modules with the A -valued scalar product (cid:104)⊕ i c i , ⊕ i c (cid:48) i (cid:105) := (cid:88) i ∈ I (cid:104) c i , c (cid:48) i (cid:105) i , where (cid:104)− , −(cid:105) i is the A -valued scalar product on C i . We then let C be the closure of C alg with respect to the norm induced by this scalar product. The scalar product extendsby continuity and equips C with the structure of an A -Hilbert C ∗ -module. We have anobvious mutual orthogonal family ( e i ) i ∈ I of isometries e i : C i → C . Note that for c in C we have (cid:107) c (cid:107) = (cid:88) i ∈ I (cid:107) e ∗ i ( c ) (cid:107) i . (3.13)The pair ( C, ( e i ) i ∈ I ) represents the classical notion of an orthonormal sum of the family( C i ) i ∈ I in Hilb ( A ). We claim that ( C, ( e i ) i ∈ I ) is an orthogonal sum of the family ( C i ) i ∈ I in the sense of Definition 3.16. To this end we must show that the associated multipliermaps (3.9) and (3.10) are isomorphisms.We first show that both transformations are injective. In view of the presence of theinvolution it suffices to consider only right multipliers. Let f be in Hom ( D, C ) and let R ( f )denote its associated right multiplier. We assume that R ( f ) = 0. Then 0 = R ( f )( e ∗ i ) = e ∗ i f for all i in I . For every element d in D we get, using (3.13), (cid:107) f ( d ) (cid:107) = (cid:88) i ∈ I (cid:107) e ∗ i ( f ( d )) (cid:107) i = 0 . This implies f = 0.We now show that the associated multiplier maps (3.9) and (3.10) are surjective. Let L = ( L D (cid:48) ) D (cid:48) ∈ Ob(
Hilb ( A )) be in LM( C, D ). We define a morphism f : C → D which is apotential preimage of L under the associated left multiplier map. We first construct f as a bounded linear map C → D . Let c be in C . It gives rise to a uniquely determinedmorphism h c : A → C such that h c (1 A ) = c . We shall see that this morphism belongs to K ( A, C ). This is clear if c is in C alg . We then use that the map c (cid:55)→ h c is a continuousmap from C to Hom ( A, C ), that K ( A, C ) is a closed subspace of
Hom ( A, C ), and that C alg is dense in C . We then set f ( c ) := L A ( h c )(1 A )22n D . One checks that this defines a linear map f : C → D . We now show that f isbounded: (cid:107) f ( c ) (cid:107) = (cid:107) L A ( h c )(1 A ) (cid:107) ≤ (cid:107) L A ( h c ) (cid:107) ≤ (cid:107) L A (cid:107)(cid:107) h c (cid:107) = (cid:107) L A (cid:107)(cid:107) c (cid:107) . In order to show that f admits an adjoint we first consider a similar construction for rightmultipliers. Let R be in RM( D, C ). Then we construct a morphism f (cid:48) : D → C whichis a potential preimage of R under the associated right multiplier map. We again firstconstruct f (cid:48) as a bounded linear map D → C . Let d be in D . We claim that the sum (cid:80) i ∈ I e i ( R C i ( e ∗ i )( d )) converges in C . To this end we observe that for every finite subset J of I (cid:88) j ∈ J (cid:107) e j ( R C j ( e ∗ j )( d )) (cid:107) = (cid:107) (cid:88) j ∈ J e j ( R C j ( e ∗ j )( d )) (cid:107) = (cid:107) R C ( (cid:88) j ∈ J e j e ∗ j )( d ) (cid:107) ≤ (cid:107) R C (cid:107) (cid:107) (cid:88) j ∈ J e j e ∗ j (cid:107) (cid:107) d (cid:107) ≤ (cid:107) R (cid:107) (cid:107) d (cid:107) . For the last step we used that (cid:80) j ∈ J e j e ∗ j is an orthogonal projection and therefore boundedby 1 independently of J . We now define f (cid:48) : D → C by f (cid:48) ( d ) := (cid:88) i ∈ I e i ( R C i ( e ∗ i )( d )) . The estimate above shows that f (cid:48) is bounded by (cid:107) R (cid:107) .We now apply this construction for right multipliers to R := L ∗ . In this case we show thatthe adjoint of f is f (cid:48) and vice versa. It suffices to show that for all c in C alg and d in D we have the equality (cid:104) f ( c ) , d (cid:105) = (cid:104) c, f (cid:48) ( d ) (cid:105) . Indeed, for any sufficiently large finite subset J of I we have (cid:104) c, f (cid:48) ( d ) (cid:105) = (cid:104) c, (cid:88) i ∈ I e i ( R C i ( e ∗ i )( d )) (cid:105) = (cid:88) i ∈ J (cid:104) c, e i ( R C i ( e ∗ i )( d )) (cid:105) = (cid:88) i ∈ J (cid:104) e ∗ i ( c ) , R C i ( e ∗ i )( d ) (cid:105) i = (cid:88) i ∈ J (cid:104) L C i ( e i )( e ∗ i ( c )) , d (cid:105) D = (cid:88) i ∈ J (cid:104) L C ( e i e ∗ i )( c ) , d (cid:105) D = (cid:88) i ∈ J (cid:104) L A ( e i e ∗ i h c )(1 A ) , d (cid:105) D = (cid:104) L A ( (cid:88) i ∈ J e i e ∗ i h c )(1 A ) , d (cid:105) D = (cid:104) L A ( h c )(1 A ) , d (cid:105) D = (cid:104) f ( c ) , d (cid:105) D . We now show that the associated right multiplier of f (cid:48) is R . Indeed, for every j in I andmorphism g : C j → D (cid:48) we have g ( e ∗ j ( f (cid:48) ( d )) = (cid:88) i ∈ I g ( e ∗ j ( e i ( R C i ( e ∗ i )( d )))) = R D (cid:48) ( ge ∗ j )( d ) . R is given by right-composition by f (cid:48) . Using the involution we concludethat the associated left multiplier of f is L .This finishes the proof of the surjectivity of the associated multiplier maps (3.9) and(3.10). Remark 3.24.
The category
Hilb ( A ) is α -additive for every very small cardinal α . In this section we explain methods to produce morphisms into or out of an orthogonal sum.This will be used in Proposition 4.3 to provide an alternative characterization of orthogonalsums. We then consider Yoneda type embeddings of C ∗ -categories into categories of Hilbert C ∗ -modules and use this to provide a third characterization of orthogonal sums in termsof orthogonal sums of Hilbert C ∗ -modules. And finally, we provide some technical resultspreparing [BE]. Given a family of objects indexed by a set I we can consider a partition( J k ) k ∈ K of I . In this case we can consider the sum over K over the family of partial sumsover the subfamilies indexed by the subsets J k . We will show that the result is isomorphicto the total sum over I .Let C be in C ∗ Cat , let ( C i ) i ∈ I be a family of objects of C , and assume that ( C i ) i ∈ I admitsan orthogonal sum ( C, ( e i ) i ∈ I ). Let D be an object of C , and let ( h i ) i ∈ I and ( h (cid:48) i ) i ∈ I befamilies of morphisms h i : D → C i and h (cid:48) i : C i → D . Corollary 4.1.
1. There exists a morphism h : D → C (often denoted by (cid:80) i ∈ I e i h i ) with e ∗ j h = h j forall j in I if and only if ( h i ) i ∈ I is square summable.If ( h i ) i ∈ I is square summable, then h is uniquely determined.2. There exists a morphism h (cid:48) : C → D (often denoted by (cid:80) i ∈ I h (cid:48) i e ∗ i ) with h (cid:48) e j = h (cid:48) j forall j in I if and only if ( h (cid:48) i ) i ∈ I is square summable.If ( h (cid:48) i ) i ∈ I is square summable, then h (cid:48) is uniquely determined.Proof. By Lemma 3.14 we obtain multipliers corresponding to h or h (cid:48) if and only if thecorresponding families are square summable. In view of the definition of an orthogonal sumthese multipliers lift uniquely to the desired morphisms under the associated multipliermorphism maps (3.9) or (3.10), respectively.24he following corollary states that a map into an orthogonal sum, or a map out of anorthogonal sum, respectively, is uniquely determined by its compositions with the structuremaps of the sum. We keep the notation introduced before Corollary 4.1. We considerpairs of morphisms f, f (cid:48) : D → C , k, k (cid:48) : C → D , and g, g (cid:48) : C → C . Corollary 4.2.
1. If e ∗ j f = e ∗ j f (cid:48) for all j in I , then f = f (cid:48) .2. If ke j = k (cid:48) e j for all j in I , then k = k (cid:48) .3. If e ∗ i ge j = e ∗ i g (cid:48) e j for all i, j in I , then g = g (cid:48) .Proof. Assertions 1 and 2 immediately follow from the uniqueness statements in Corol-lary 4.1.We show Assertion 3. Fixing j in I , the uniqueness statement in Corollary 4.1.1 (applied tothe family of morphisms ( h i ) i ∈ I : C j → C i defined by h i := e ∗ i ge j for all i in I ) implies that ge j = g (cid:48) e j . Then the uniqueness statement in Corollary 4.1.2 (applied to h (cid:48) i := ge i : C i → C for every i in I ) implies that g = g (cid:48) .The following proposition provides an alternative characterization of orthogonal sums interms of morphisms. Let C be in C ∗ Cat , let ( C i ) i ∈ I be a family of objects of C , and let C be an object of C with a family of mutually orthogonal isometries ( e i ) i ∈ I with e i : C i → C for every i in I . Proposition 4.3. ( C, ( e i ) i ∈ I ) is an orthogonal sum in C of the family ( C i ) i ∈ I if and onlyif the following two conditions are satisfied:
1. For every object D of C and every square summable family ( h i ) i ∈ I of morphisms h i : D → C i there exists a unique morphism h : D → C with e ∗ i h = h i for all i in I .2. For every object D of C and every square summable family ( h (cid:48) i ) i ∈ I of morphisms h (cid:48) i : C i → D there exists a unique morphism h (cid:48) : C → D with h (cid:48) e i = h (cid:48) i for all i in I .Proof. Let ( C, ( e i ) i ∈ I ) be the orthogonal sum. Then Conditions 1 and 2 are satisfied byCorollary 4.1.To prove the converse we assume Conditions 1 and 2. We have to show that the associatedmultiplier maps (3.9) and (3.10) are isomorphisms. We consider only the case of the Note that by using the involution we see that Condition 1 is actually equivalent to Condition 2. D of C and let R be in RM( D, C ). We define a family ofmorphisms ( h i ) i ∈ I with h i : D → C i for every i in I by setting h i := R C i ( e ∗ i ). By Lemma3.14(1) we see that the family ( h i ) i ∈ I is square summable. Condition 1 then implies theexistence of a unique morphism h : D → C whose associated right multiplier is R . Thisshows that the associated right multiplier map (3.9) is bijective.Assume that D is a full subcategory of C in C ∗ Cat . Assume that ( C i ) i ∈ I is a family ofobjects in D and ( C, ( e i ) i ∈ I ) is an object of D together with a mutually orthogonal familyof isometries e i : C i → C . Corollary 4.4. If ( C, ( e i ) i ∈ I ) represents an orthogonal sum of the family ( C i ) i ∈ I in C ,then it also represents an orthogonal sum of this family in D .Proof. This immediately follows from the characterization of orthogonal sums given in theCorollary 4.2 which only involves conditions formulated in the language of D .Note that a verification of this corollary by checking the conditions in Definition 3.16directly seems to be much more tricky since one would have to extend multipliers in D tomultipliers in C .In the following we associate to every unital C ∗ -category C a C ∗ -algebra A ( C ) andconstruct a Yoneda type embedding M : C → Hilb ( A ( C )), where Hilb ( A ( C )) is the C ∗ -category of right Hilbert A ( C )-modules. We will then see by using Corollary 4.4 thatthis embedding preserves orthogonal sums. So orthogonal sums in C can be completelyunderstood in terms of orthogonal sums in Hilb ( A ( C )). Let C ∗ Cat nu i denote the wide subcategory of C ∗ Cat nu of functors which are injective onobjects. We consider the functor A : C ∗ Cat nu i → C ∗ Alg nu (4.1)defined in [Joa03, Def. 2], see also [Bun, Def. 6.5]. Remark 4.5.
For the sake of self-containedness we recall the definition of the functor A .Let C be in C ∗ Cat nu i . We start with the description of a ∗ -algebra A alg ( C ). The underlying C -vector space of A alg ( C ) is the algebraic direct sum A alg ( C ) := (cid:77) C,C (cid:48) ∈ Ob( C ) Hom C ( C, C (cid:48) ) . (4.2) We thank Ch. Voigt for suggesting this idea.
26 morphism f in C gives rise to an element in A alg ( C ) which will also be denoted by f (or also by f [ C (cid:48) , C ]). The product ( g, f ) (cid:55)→ gf on A alg ( C ) is defined by gf := (cid:40) g ◦ f if g and f are composable ,0 otherwise . (4.3)The ∗ -operation on C induces an involution on A alg ( C ) in the obvious way. One can checkthat A alg ( C ) is a pre- C ∗ -algebra. We equip A alg ( C ) with the maximal C ∗ -norm and define A ( C ) as the closure of A alg ( C ). We have a natural transformation id → A : C ∗ Cat nu i → C ∗ Alg nu . (4.4)Its evaluation on C is the morphism C → A ( C ) which sends all objects of C to the uniqueobject of A ( C ) (we consider the C ∗ -algebra as a C ∗ -category with a single object), andevery morphism f in C to the corresponding element of A ( C ) denoted by the same symbol.Note that the assignment C (cid:55)→ A ( C ) is not a functor on C ∗ Cat nu since non-composablemorphisms in a C ∗ -category may become composable after applying a functor to another C ∗ -category which is incompatible with the product described in (4.3).Let C be in C ∗ Cat . We apply the functor A from (4.1) and get A ( C ) in C ∗ Alg nu . Thenwe get a new (large) unital C ∗ -category Hilb ( A ( C )) of right Hilbert A ( C )-modules. Weconsider A ( C ) as an object of Hilb ( A ( C )) in the natural way. A morphism f : C → C (cid:48) in A alg ( C ) gives rise to the matrix f [ C (cid:48) , C ] in A ( C ) with the single non-trivial entry f atposition ( C (cid:48) , C ). The element 1 C [ C, C ] is an orthogonal projection in A ( C ). Its image isa submodule M C := 1 C [ C, C ] A ( C ) . (4.5)A morphism f : C → C (cid:48) induces an A ( C )-linear map M f : M C → M C (cid:48) given by leftmultiplication by f [ C (cid:48) , C ]. One checks that this map has an adjoint given by M ∗ f = M f ∗ .In this way we get a functor between C ∗ -categories M : C → Hilb ( A ( C )) . (4.6) Lemma 4.6.
The functor M is fully faithful.Proof. We first show that M is faithful. Let f : C → C (cid:48) be a morphism in C . If M f = 0,then 0 = M f (1 C [ C, C ]). This implies f [ C (cid:48) , C ] = 0 in M C (cid:48) . This finally implies that f = 0since the maps 1 C (cid:48) A alg ( C ) → M C (cid:48) → A ( C ) are both injective.We now show that M is full. Let F : M C → M C (cid:48) be a morphism in Hilb ( A ( C )). Then weget F (1 C [ C, C ]) in M C (cid:48) . For every C (cid:48)(cid:48) in C we have by A ( C )-linearity F ( id C [ C, C ]1 C (cid:48)(cid:48) [ C (cid:48)(cid:48) , C (cid:48)(cid:48) ]) = F ( id C [ C, C ])1 C (cid:48)(cid:48) [ C (cid:48)(cid:48) , C (cid:48)(cid:48) ] . The left hand side vanishes if C (cid:48)(cid:48) (cid:54) = C and gives F ( id C [ C, C ]) in the case C = C (cid:48)(cid:48) . Thismeans that F ( id C [ C, C ]) = f [ C (cid:48) , C ] for some f : C → C (cid:48) . We conclude that M f = F .27he functor M identifies C with a full subcategory of Hilb ( A ( C )). Since Hilb ( A ( C ))admits sums for all small families we conclude by Corollary 4.4 that this embedding pre-serves orthogonal sums. By Lemma 3.23 orthogonal sums in the target can be understoodin the classical sense. In the following we give a precise statement.Assume that ( C i ) i ∈ I is a family of objects in C and that ( C, ( e i ) i ∈ I ) is an orthogonalsum of this family in C . Then we have the family ( M C i ) i ∈ I in Hilb ( A ( C )) and the pair( M C , ( M e i ) i ∈ I ). Corollary 4.7. ( M C , ( M e i ) i ∈ I ) is an orthogonal sum of ( M C i ) i ∈ I in Hilb ( A ( C )) . Assume again that ( C i ) i ∈ I is a family of objects in C . Corollary 4.8.
The family ( C i ) i ∈ I admits an orthogonal sum in C if and only if thereexists an object C in C such that M C ∼ = (cid:76) i ∈ I M C i in Hilb ( A ( C )) .Proof. If ( C, ( e i ) i ∈ I ) is an orthogonal sum of the family in C , then M C ∼ = (cid:76) i ∈ I M C i byCorollary 4.7.Conversely, if there exists C with an isomorphism M C ∼ = (cid:76) i ∈ I M C i , then since M isfully faithful we can find morphisms e i : C i → C for all i in I such that M e i is the map M C i → (cid:76) i ∈ I M C i ∼ = M C , where the first map is the canonical embedding. Then ( C, ( e i ) i ∈ I )represents the orthogonal sum of the family by Corollary 4.4. Corollary 4.9.
Any small unital C ∗ -category admits an orthogonal sum preserving em-bedding into a large C ∗ -category admitting orthogonal sums for all small families. We now present two illustrative examples of orthogonal sums of an infinite family ofobjects.
Example 4.10.
Let X be a countably infinite set. We define a C ∗ -category X as follows:1. objects: The set of objects of X is the set X ∪ { X } .2. morphisms: The morphism spaces are defined as subspaces of B ( (cid:96) ( X )). For everytwo subsets Y, Y (cid:48) of X we can consider B ( (cid:96) ( Y ) , (cid:96) ( Y (cid:48) )) as a block subspace of B ( (cid:96) ( X )) in the natural way.a) For x in X the algebra End X ( x ) is the subalgebra B ( (cid:96) ( { x } )). It is isomorphicto C . 28) For x, x (cid:48) in X with x (cid:54) = x (cid:48) we set Hom X ( x, x (cid:48) ) := 0.c) The algebra End X ( X ) is the subalgebra of diagonal operators in B ( (cid:96) ( X )). Itis isomorphic to (cid:96) ∞ ( X ).d) For x in X we let Hom X ( x, X ) be the subspace of B ( (cid:96) ( { x } ) , (cid:96) ( X )) generatedby the canonical inclusion e x . Similarly we let Hom X ( X, x ) be the subspace of B ( (cid:96) ( X ) , (cid:96) ( { x } )) generated by e ∗ x . These spaces are one-dimensional.3. The composition and the involution of X is induced from B ( (cid:96) ( X )).We claim that ( X, ( e x ) x ∈ X ) is the orthogonal sum in X of the family ( x ) x ∈ X . In orderto show the claim we use Proposition 4.3. We consider only one of the four cases. Theremaining are left as an exercise. Let ( h (cid:48) x ) x ∈ X be a square summable family of morphisms h (cid:48) x : x → X . Then for every x in X we have h (cid:48) x = λ (cid:48) x e x for some uniquely determined λ (cid:48) x in C . Furthermore (cid:107) (cid:80) x ∈ J h (cid:48) x h (cid:48) , ∗ x (cid:107) = (cid:107) (cid:80) x ∈ J | λ (cid:48) x | e x e ∗ x (cid:107) = max x ∈ J | λ (cid:48) x | for every finite family J of X . Since ( h (cid:48) x ) x ∈ X is square summable it follows that ( λ (cid:48) x ) x ∈ X is uniformly bounded.The unique morphism h (cid:48) : X → X with h (cid:48) e x = h (cid:48) x for all x in X is then given by thediagonal operator on (cid:96) ( X ) given by multiplication by the bounded function x (cid:55)→ λ (cid:48) x . Example 4.11.
Let X be a countably infinite set. We define a C ∗ -category X (cid:48) as follows:1. objects: The set of objects of X (cid:48) is the set X ∪ { X } .2. morphisms: As in Example 4.10 the morphism spaces are defined as subspaces of B ( (cid:96) ( X ∪ { X } )).a) For x, x (cid:48) in X we set Hom X (cid:48) ( x, x (cid:48) ) as B ( (cid:96) ( { x } ) , (cid:96) ( { x (cid:48) } )). It is one-dimensional.b) For any x in X we set Hom X (cid:48) ( x, X ) := B ( (cid:96) ( { x } ) , (cid:96) ( X )) and Hom X (cid:48) ( X, x ) := B ( (cid:96) ( X ) , (cid:96) ( { x } )).c) End X (cid:48) ( X ) := B ( (cid:96) ( X )).3. The composition and the involution of X (cid:48) is induced from B ( (cid:96) ( X )).For every x in X we denote by e x the canonical inclusion in B ( (cid:96) ( { x } ) , (cid:96) ( X )). We claimthat ( X, ( e x ) x ∈ X ) is the orthogonal sum in X (cid:48) of the family ( x ) x ∈ X . In order to show thisclaim we use Corollary 4.4. We consider X (cid:48) as a full subcategory of Hilb ( C ) in the obviousmanner. Since ( (cid:96) ( X ) , ( e x ) x ∈ X ) is an orthogonal sum of ( x ) x ∈ X in Hilb ( C ) (in the senseof Definition 3.16) by Lemma 3.23 it is also an orthogonal sum of this family in X (cid:48) .Let ( C i ) i ∈ I be a family of objects in C and assume that it admits an orthogonal sum( (cid:76) i ∈ I C i , ( e i ) i ∈ I ). Let J be a subset of I . Then we consider the family ( e ∗ j ) j ∈ J of morphisms29 ∗ j : (cid:76) i ∈ I C i → C j . If we extend this family by zero to a family indexed by I , then byCorollary 4.1.1 applied with D := (cid:76) i ∈ I C i we can form p := (cid:80) j ∈ J e j e ∗ j in End C ( (cid:76) i ∈ I C i ).Note that pe j = e j for all j in J and pe i = 0 for i in I \ J . Lemma 4.12. p is a projection.2. If p is effective and ( E, u ) presents an image of p (Definition 2.13), then ( E, ( u ∗ e i ) i ∈ J ) represents the sum of the subfamily ( C j ) j ∈ J .Proof. We consider f in Hom fin C ( D (cid:48) , (cid:76) i ∈ I C i ). Then for a sufficiently large finite subset J (cid:48) of J (depending on f ) we have, using that ( e i ) i ∈ I is a mutual orthogonal family of isometries, p f = (cid:88) j ∈ J e j e ∗ j (cid:0) (cid:88) i ∈ J e i e ∗ i f (cid:1) = (cid:88) j ∈ J e j e ∗ j (cid:0) (cid:88) i ∈ J (cid:48) e i e ∗ i f (cid:1) = (cid:88) j ∈ J (cid:48) e j e ∗ j f = pf . We conclude that p = p . We verify similarly that p ∗ = p .We now assume that p is effective, and that u : E → (cid:76) i ∈ I C i presents an image of p . Thefamily ( u ∗ e i ) i ∈ J of morphisms u ∗ e i : C i → E is a mutually orthogonal family of isometries.We now consider the left and right multipliers for ( E, ( u ∗ e i ) i ∈ J ). We must show that theassociated multiplier morphisms m E,RD : Hom C ( D, E ) → RM(
D, E ) , m E,LD : Hom C ( E, D ) → LM(
E, D )are isomorphisms for all D in C , where we added a superscript E in order to indicate thedependence on E . It suffices to consider the case of m E,RD . The other case then follows byapplying the involution.We first show surjectivity. Let R := ( R D (cid:48) ) D (cid:48) ∈ Ob( C ) be in RM( D, E ). Pre-composition with u induces a map − ◦ u : Hom fin C (( (cid:77) i ∈ I C i , ( e i ) i ∈ I ) , D (cid:48) ) → Hom fin C (( E, ( u ∗ e i ) i ∈ J ) , D (cid:48) )and therefore extends by continuity to a map − ◦ u : K (( (cid:77) i ∈ I C i , ( e i ) i ∈ I ) , D (cid:48) ) → K (( E, ( u ∗ e i ) i ∈ J ) , D (cid:48) ) . Then Ru := ( R D (cid:48) ◦ ( −◦ u )) D (cid:48) ∈ Ob( C ) belongs to RM( D, (cid:76) i ∈ I C i ). Hence there exists a uniqelydetermined morphism f : D → (cid:76) i ∈ I C i such that m C,RD ( f ) = Ru . Then m E,RD ( u ∗ f ) = R .Assume now that f : D → E is a morphism such that m E,RD ( f ) = 0. This means that hf = 0 for all objects D (cid:48) and generators h of K ( E, D (cid:48) ). Note that e ∗ i u is a generator of K ( E, C i ). Hence in particular we have e ∗ i uf = 0 for all i in I . This implies uf = 0 andtherefore f = u ∗ uf = 0. 30he following results fit into the present discussion but will only be used in the follow uppaper [BE]. Let C be in C ∗ Cat . Let ( C i ) i ∈ I be a family of objects in C and ( C, ( e i ) i ∈ I )be an orthogonal sum of the family. Let furthermore ( J k ) k ∈ K be a partition of the set I .For every k in K we can form the projection p k := (cid:80) i ∈ J k e i e ∗ i by Lemma 4.12. Lemma 4.13.
Assume that for any k in K the projection p k is effective with image ( E k , u k ) .Then the sum of the family ( E k ) k ∈ K exists and is represented by ( (cid:76) i ∈ I C i , ( u k ) k ∈ K ) .Proof. Since the members of the family ( J k ) k ∈ K are mutually disjoint we have p k p k (cid:48) = 0for all k, k (cid:48) in K with k (cid:54) = k . This implies that ( u k ) k ∈ K is a mutually orthogonal family ofisometries.We must show that the associated multiplier morphisms (3.9) and (3.10) m (cid:48) ,RD : Hom C ( D, (cid:77) i ∈ I C i ) → RM( D, ( (cid:77) i ∈ I C i , ( u k ) k ∈ K )) ,m (cid:48) ,LD : Hom C ( (cid:77) i ∈ I C i , D ) → LM(( (cid:77) i ∈ I C i , ( u k ) k ∈ K ) , D )are isomorphisms for all objects D in C . Here the superscript (cid:48) is added in order todistinguish these maps from the associated multiplier maps m RD and m LD of ( C, ( e i ) i ∈ I ).We again consider the case of right multipliers. The case of left multipliers then follows byapplying the involution. We have inclusions K (( (cid:77) i ∈ I C i , ( e i ) i ∈ I ) , D (cid:48) ) ⊆ K (( (cid:77) i ∈ I C i , ( u k ) k ∈ K ) , D (cid:48) )for all D (cid:48) in C . Hence we have a restriction map ! fitting into the diagram Hom C ( D, (cid:76) i ∈ I C i ) m (cid:48) ,RD (cid:116) (cid:116) m RD ∼ = (cid:42) (cid:42) RM( D, ( (cid:76) i ∈ I C i , ( u k ) k ∈ K )) ! (cid:47) (cid:47) RM( D, ( (cid:76) i ∈ I C i , ( e i ) i ∈ I ))This already implies that the map marked by m (cid:48) ,RD is injective.We will now show that ! is injective. To this end we assume that R = ( R D (cid:48) ) D (cid:48) ∈ Ob( C ) is inRM( D, ( (cid:76) i ∈ I C i , ( u k ) k ∈ K )) and is sent to zero by !. We must show that R E k ( u ∗ k ) = 0 forall k in K . We have for all i in J k that e ∗ i u k R E k ( u ∗ k ) = R C i ( e ∗ i u k u ∗ k ) = R C i ( e ∗ i p k ) = R C i ( e ∗ i ) = 0 , where for the last equality we use the assumption on R . This implies by Lemma 4.12.2and the uniqueness assertion in Corollary 4.1.1 (applied to the sum ( E k , ( e i u ∗ k ) i ∈ J k ) andthe family of morphisms ( e ∗ i u k R E k ( u ∗ k )) i ∈ J k ) that R E k ( u ∗ k ) = 0.The injectivity of ! implies that m (cid:48) ,RD is surjective by a diagram chase.31 xample 4.14. Let ( C i ) i ∈ I and ( C (cid:48) i ) i ∈ I be two families of objects in C with the same indexset. We assume that they admit orthogonal sums ( (cid:76) i ∈ I C i , ( e i ) i ∈ I ) and ( (cid:76) i ∈ I C (cid:48) i , ( e (cid:48) i ) i ∈ I ).Let ( f i ) i ∈ I be a family of morphisms f i : C i → C (cid:48) i . We assume that sup i ∈ I (cid:107) f i (cid:107) < ∞ . UsingCorollary 4.1.1 applied to ( f i e ∗ i ) i ∈ I we get a unique morphism f : (cid:76) i ∈ I C i → (cid:76) i ∈ I C (cid:48) i suchthat e (cid:48) , ∗ j f = f j e ∗ j for all j in I . On the other hand, using Corollary 4.1.2 applied to thefamily ( e (cid:48) i f i ) i ∈ I we get a unique morphism f (cid:48) : (cid:76) i ∈ I C i → (cid:76) i ∈ I C (cid:48) i satisfying f (cid:48) e j = e (cid:48) j f j for all j in I .We claim that f = f (cid:48) . For all i, j in I with i (cid:54) = j we have e (cid:48) , ∗ i f e j = 0 = e (cid:48) , ∗ i f (cid:48) e j . Furthermore e (cid:48) , ∗ i f e i = f i e ∗ i e i = f i = e (cid:48) , ∗ i e (cid:48) i f i = e (cid:48) , ∗ i f (cid:48) e i . This first implies that e (cid:48) , ∗ i ( f − f (cid:48) ) e j = 0 for all i, j in I , which in turn implies f = f (cid:48) .We will usually use the suggestive notation ⊕ i ∈ I f i : (cid:77) i ∈ I C i → (cid:77) i ∈ I C (cid:48) i (4.7)for the morphism f (or equivalently, f (cid:48) ) considered above.Let C be in C ∗ Cat , C be an object of C , and ( p i ) i ∈ I be a mutually orthogonal, commutingfamily of projections on C . Definition 4.15.
We say that C is the orthogonal sum of the images of the family if thefollowing are satisfied:1. For every i in I the projection p i is effective (see Definition 2.15).2. If ( D i , u i ) is a choice of an image of p i for every i in I (see Definition 2.13), then ( C, ( u i ) i ∈ I ) represents the orthogonal sum of the family ( D i ) i ∈ I . Note that the validity of the conditions in Definition 4.15 does not depend on the choicesinvolved in the images and the direct sum.
In this section we compare our approach to orthogonal sums in C ∗ -categories with theversion of Antoun and Voigt. Since our notion of an orthogonal sum differs from thatin this reference, in this section we take the chance to recall their versions in detail andexplain the similarities and differences. The preprint [AV] appeared on the arXiv shortly before we finished writing this paper. For earlier andsimilar approaches see also Kandelaki [Kan01] and Vasselli [Vas07].
32e start with recalling the notion of a multiplier morphism and the multiplier categoryM C [AV, Sec. 2]. For the sake of self-containedness we provide complete proofs. Let C bein C ∗ Cat nu , and let C, D be objects of C . Definition 5.1.
1. The Banach space of left multiplier morphisms from C to D is the Banach space ofuniformly bounded natural transformations of Ban -valued functors
Hom C ( − , C ) → Hom C ( − , D ) on C op (see Definition 3.10).2. The Banach space of right multiplier morphisms from C to D is the Banach space ofuniformly bounded natural transformations of Ban -valued functors
Hom C ( D, − ) → Hom C ( C, − ) on C .3. A multiplier morphism from C to D is a pair ( L, R ) of a left and a right multipliermorphism from C to D such that for every f in Hom C ( F, C ) and every g in Hom C ( D, E ) we have gL F ( f ) = R E ( g ) f . (5.1) We write
MHom C ( C, D ) for the C -vector space of multiplier morphisms from C to D . In the following we spell out this definition in detail and explain the notation appearing in(5.1). A left multiplier morphism L : C → D is a uniformly bounded family ( L E ) E ∈ Ob( C ) of C -linear maps L E : Hom C ( E, C ) → Hom C ( E, D ) such that for every h in Hom C ( E, C ) andevery g in Hom C ( F, E ) we have L F ( hg ) = L E ( h ) g .Similarly, a right multiplier morphism R : C → D is given by a uniformly bounded family( R E ) E ∈ Ob( C ) of C -linear maps R E : Hom C ( D, E ) → Hom C ( C, E ) such that for every h in Hom C ( D, E ) and every g in Hom C ( E, F ) we have R F ( gh ) = gR E ( h ).Below, in order to simplify the notation, we will omit the subscripts and write L ( h ) insteadof L E ( h ) or R ( g ) instead of R E ( g ).Let C, D, E be objects of C , let ( L, R ) be in
MHom C ( C, D ), and ( L (cid:48) , R (cid:48) ) be in MHom C ( D, E ).Then the pair of compositions ( L (cid:48) L, RR (cid:48) ) belongs to
MHom C ( C, E ). In this way we get a C -bilinear and associative law of composition of multiplier morphisms MHom C ( C, D ) × MHom C ( D, E ) → MHom C ( C, E ) . (5.2)For every object C in C we have an identity multiplier morphism id C in MHom C ( C, C ).Finally, the involution of C induces an anti-linear involution( − ) ∗ : MHom C ( C, D ) → MHom C ( D, C ) , ( L, R ) ∗ := ( R ∗ , L ∗ ) .
33n detail, if L = ( L E ) E ∈ Ob( C ) and R = ( R E ) E ∈ Ob( C ) , then L ∗ = ( L ∗ E ) E ∈ Ob( C ) with L ∗ E ( f ) := R E ( f ∗ ) ∗ for every f in Hom C ( E, D ), and analogously R ∗ = ( R ∗ E ) E ∈ Ob( C ) with R E ( f ) = L E ( f ∗ ) ∗ for every f in Hom C ( C, E ).The multiplier category M C of C is the object of ∗ Cat C defined as follows. Definition 5.2.
1. The objects of M C are the objects of C .2. The C -vector space of morphisms in M C from C to D is the space of multipliermorphisms MHom C ( C, D ) .3. The composition and the involution are defined as described above. Let C be in C ∗ Cat nu . Proposition 5.3. M C belongs to C ∗ Cat .Proof.
We will show that the norm of the Banach space of multiplier morphisms exhibitsM C as a C ∗ -category. We define the norm a multiplier morphism M = ( L, R ) by (cid:107) M (cid:107) := max {(cid:107) L (cid:107) , (cid:107) R (cid:107)} , (5.3)see Definition 3.10. The involution ∗ on M C is then isometric.Next we show that actually (cid:107) L (cid:107) = (cid:107) R (cid:107) . The argument is similar to the argument fordouble centralizers of C ∗ -algebras; cf. [Mur90, Lem. 2.1.4].First of all note that for a morphism f : C → D in a C ∗ -category we have (cid:107) f (cid:107) = sup g (cid:107) f g (cid:107) = sup h (cid:107) hf (cid:107) , (5.4)where g runs over all morphisms with target C and (cid:107) g (cid:107) ≤
1, and h runs over all morphismswith domain D and (cid:107) h (cid:107) ≤
1: In fact, assume that f (cid:54) = 0. Then we have (cid:107) f (cid:107) ≥ sup g (cid:107) f g (cid:107) ≥ (cid:13)(cid:13)(cid:13) f f ∗ (cid:107) f (cid:107) (cid:13)(cid:13)(cid:13) = (cid:107) f ∗ f (cid:107)(cid:107) f (cid:107) − = (cid:107) f (cid:107) (cid:107) f (cid:107) − = (cid:107) f (cid:107) , where the first inequality follows from the sub-multiplicativity of the norm, while thesecond inequality follows from specializing at g = f ∗ / (cid:107) f (cid:107) .34ssume that M = ( L, R ) is a multiplier morphism from C to D . Then we have (cid:107) L (cid:107) = sup f (cid:107) L ( f ) (cid:107) (5.4) = sup h sup f (cid:107) hL ( f ) (cid:107) (5.1) = sup h sup f (cid:107) R ( h ) f (cid:107) (5.4) = sup h (cid:107) R ( h ) (cid:107) = (cid:107) R (cid:107) , where f runs over all morphisms with target C and (cid:107) f (cid:107) ≤
1, and h runs over all morphismswith domain D and (cid:107) h (cid:107) ≤ C is complete since the spaces of left and right multipliers are complete.It is furthermore easy to see that the norm is sub-multiplicative for compositions.It remains to verify the strong C ∗ -inequality (2.2). We consider a multiplier morphisms M = ( L , R ) from C to E and a multiplier morphism M = ( L , R ) from C to D . Wethen have (cid:107) M ∗ M + M ∗ M (cid:107) = (cid:107) R ∗ L + R ∗ L (cid:107) = sup f (cid:107) R ∗ ( L ( f )) + R ∗ ( L ( f )) (cid:107)≥ sup f (cid:107) f ∗ R ∗ ( L ( f )) + f ∗ R ∗ ( L ( f )) (cid:107) (5.1) = sup f (cid:107) L ∗ ( f ∗ ) L ( f ) + L ∗ ( f ∗ ) L ( f ) (cid:107) = sup f (cid:107) L ( f ) ∗ L ( f ) + L ( f ) ∗ L ( f ) (cid:107) ! ≥ sup f (cid:107) L ( f ) ∗ L ( f ) (cid:107) = sup f (cid:107) L ( f ) (cid:107) = (cid:107) M (cid:107) = (cid:107) M ∗ M (cid:107) , where the supremum runs over all f with target C and (cid:107) f (cid:107) ≤
1, and at the markedinequality we used the C ∗ -inequality of C .Every morphism f in Hom C ( C, D ) naturally defines a multiplier morphism M f := ( L f , R f )in MHom C ( C, D ), where L f = f ◦ − and R f = − ◦ f . We have a morphism C → M C in ∗ Cat nu C which is the identity on the objects and given by f (cid:55)→ ( L f , R f ) on morphisms. Lemma 5.4.
The morphism C → M C is the inclusion of a closed ideal.Proof. For f in Hom C ( C, D ) we have (cid:107) M f (cid:107) = (cid:107) L f (cid:107) (5.4) = (cid:107) f (cid:107) . This implies that C → M C is an isometric inclusion and therefore C is closed in M C .Consider a multiplier M = ( L, R ) from C to D in C . Let h : D → E be given. Then M h M = ( L h L, RR h ), and we have L h ( L ( l )) = hL ( l ) = R ( h ) l = L R ( h ) ( l )35nd R ( R h ( g )) = R ( gh ) = gR ( h ) = R R ( h ) ( g ) . We conclude that M h M = M R ( h ) . Similarly we show for l : E → C that M M l = M L ( l ) .Next we argue that the algebraic conditions on multiplier morphisms alone already implythe boundedness assumptions. Let C be in C ∗ Cat nu , and let C, D be objects of C . Wedefine an algebraic left multiplier from C to D as a natural transformation of C -vectorspace valued functors Hom C ( − , C ) → Hom C ( − , D ) on C op . Similarly, an algebraic rightmultiplier from C to D is a natural transformation of C -vector space valued functors Hom C ( D, − ) → Hom C ( C, − ) on C . An algebraic multiplier morphism is a pair ( L, R ) ofan algebraic left multiplier morphism L = ( L E ) E ∈ Ob( C ) , and an algebraic right multipliermorphism R = ( R E ) E ∈ Ob( C ) such that for every f in Hom C ( F, C ) and every g in Hom C ( D, E )we have gL F ( f ) = R E ( g ) f . We let M alg Hom C ( C, D ) denote the C -vector space of algebraicmultiplier morphisms.In order to define the strict topology on multipliers we introduce the following collectionsof seminorms. For every morphism f with target C we define the seminorm l f : M alg Hom C ( C, D ) → R ≥ , l f (( L, R )) := (cid:107) L ( f ) (cid:107) . (5.5)For every morphism h with domain D we define the seminorm r h : M alg Hom C ( C, D ) → R ≥ , r h (( L, R )) := (cid:107) R ( h ) (cid:107) . (5.6)The strict topology on M alg Hom C ( C, D ) is the locally convex topology given by the familyof semi-norms ( l f ) f ∪ ( r h ) h , where f runs over morphisms with target C and h runs overmorphisms with domain D . Lemma 5.5.
1. The natural inclusion
MHom C ( C, D ) → M alg Hom C ( C, D ) is an isomorphism.2. Hom C ( C, D ) is strictly dense in MHom C ( C, D ) .3. MHom C ( C, D ) is complete with respect to the strict topology.Proof. Let (
L, R ) be an algebraic multiplier morphism from C to D . We first show thatthe members of the families L = ( L E ) E ∈ Ob( C ) and R = ( R E ) E ∈ Ob( C ) are bounded. We fixan object E of C . Then L E : Hom C ( E, C ) → Hom C ( E, D ) is a linear map of Banach spaces.We show that its graph is closed and conclude that it is continuous and hence bounded.Let ( f i ) i be a net in Hom C ( E, C ) such that lim i f i = f and assume that lim i L ( f i ) =: g exists. For every h in Hom C ( D, F ) we have (cid:107) hL ( f ) − hg (cid:107) ≤ (cid:107) hL ( f ) − hL ( f i ) (cid:107) + (cid:107) hL ( f i ) − hg (cid:107) = (cid:107) R ( h )( f − f i ) (cid:107) + (cid:107) h ( L ( f i ) − g ) (cid:107) . lim i we get (cid:107) h ( L ( f ) − g ) (cid:107) = 0 for all h . By (5.4) we can conclude that L ( f ) = g .This is a first step towards the verification of Assertion 1.We show now the Assertion 2. We actually show the stronger assertion hat Hom C ( C, D )is strictly dense in M alg Hom C ( C, D ). Let M = ( L, R ) be in M alg Hom C ( C, D ) and let ( h i ) i be a selfadjoint approximate unit of End C ( D ). We first show that lim i h i l = l for everymorphism l with target D and lim i kh i = k for every morphism with domain D . We givethe argument for the first case. Note that (cid:107) h i l − l (cid:107) = (cid:107) ( h i l − l )( h i l − l ) ∗ (cid:107) = (cid:107) ( ll ∗ − h i ll ∗ ) + ( ll ∗ − ll ∗ h i ) − ( ll ∗ − h i ll ∗ h i ) (cid:107) . We can rewrite the last term in the form h i ll ∗ h i = h i √ ll ∗ √ ll ∗ h i . Since lim i h i ll ∗ = ll ∗ = lim i ll ∗ h i and lim i h i √ ll ∗ = √ ll ∗ = lim i √ ll ∗ h i we conclude that lim i (cid:107) h i l − l (cid:107) = 0.We now come back to the Assertion 2 and note that M h i M = M R ( h i ) . We show that lim i M R ( h i ) = M in the strict topology. Let f be in Hom C ( C, D ). Then we have L R ( h i ) ( f ) = R ( h i ) f = h i L ( f )and hence lim i L R ( h i ) ( f ) = lim i h i L ( f ) = L ( f ). Similarly, for g in Hom C ( D, C ) we have lim i R L ( h i ) ( g ) = R ( g ). This proves Assertion 2.We now finish the proof of Assertion 1. If ( L, R ) is in M alg Hom C ( C, D ), then we havealready seen that L and R are implemented by families L = ( L E ) E ∈ Ob( C ) and R =( R E ) E ∈ Ob( C ) of bounded maps. It remains to show that these families are uniformlybounded. We now note that L = lim i L R ( h i ) in the strict topology, where ( h i ) i is aselfadjoint approximate unit as above. Thus L ( f ) = lim i R D ( h i ) f for all f in Hom C ( E, C ).In particular, (cid:107) L ( f ) (cid:107) ≤ sup i (cid:107) R D (cid:107)(cid:107) h i (cid:107)(cid:107) f (cid:107) ≤ (cid:107) R D (cid:107)(cid:107) f (cid:107) since sup i (cid:107) h i (cid:107) ≤
1. This showsthat (cid:107) L (cid:107) ≤ (cid:107) R D (cid:107) . Similarly one shows that (cid:107) R (cid:107) ≤ (cid:107) L D (cid:107) .We finally show Assertion 3. The arguments are the same as for double centralizers for C ∗ -algebras [Bus68, Prop. 3.6]. Let ( M ν ) ν be a Cauchy net with respect to the stricttopology in MHom C ( C, D ). Set M ν = ( L ν , R ν ). Then L := lim ν L ν and R := lim ν R ν existpointwise and obviously define an element M = ( L, R ) of M alg Hom C ( C, D ). We now useAssertion 1 in order to conclude that M belongs to MHom C ( C, D ).One can also check that the composition (5.2) is separately strictly continuous, and jointlystrictly continuous on bounded subsets.Let C be an object of C . Lemma 5.6.
Assume that C is unital.1. For every object D we have the equalities Hom C ( C, D ) =
MHom C ( C, D ) and Hom C ( D, C ) =
MHom C ( D, C ) . . The strict and norm topologies on MHom C ( C, D ) and MHom C ( D, C ) coincide.Proof. Assertion 1 is an immediate consequence of Lemma 5.4.We now show Assertion 2. It is clear that the norm on
MHom C ( C, D ) bounds (up to scale)all the seminorms l f in (5.5) and r h in (5.6). In particular, for a multiplier morphism M = ( L, R ) we have l id C ( L ) ≤ (cid:107) L (cid:107) = (cid:107) M (cid:107) . On the other hand we have (cid:107) M (cid:107) = (cid:107) R (cid:107) = sup g (cid:107) R ( g ) (cid:107) = sup g (cid:107) R ( g ) id C (cid:107) = sup g (cid:107) gL ( id C ) (cid:107) ≤ l id C ( L ) , where the supremum runs over all morphisms g with domain D and (cid:107) g (cid:107) ≤
1. This showsthat the seminorm l id C is equivalent to the norm on MHom C ( C, D ).From now on for a multiplier morphisms (
L, R ) from C to D we use the same notation f as for morphisms and write gf instead of R ( g ) and f h instead of L ( h ).Let C be in C ∗ Cat nu , let ( C i ) i ∈ I be a family of objects of C , and let C be an object of C together with a family ( e i ) i ∈ I of mutually orthogonal multiplier morphisms e i : C i → C . Definition 5.7 ([AV, Defn. 2.1]) . ( C, ( e i ) i ∈ I ) is an orthogonal AV-sum of the family ( C i ) i ∈ I if the sum (cid:80) i ∈ I e i e ∗ i converges strictly to the identity multiplier morphism of C . We use the term AV-sum (AV stands for Antoun–Voigt) in order to distinguish this notionfrom the one defined in Definition 3.16.Orthogonal AV-sums enjoy the following universal property.Let ( f i ) i ∈ I be a uniformly bounded family of morphisms f i : C i → D . Lemma 5.8.
If the sum (cid:80) i ∈ I f i e ∗ i converges strictly, then there exists a unique multipliermorphism f : C → D with f e i = f i for all i in I .Proof. The sum converges to a multiplier morphism f = ( L, R ). Since the compositionof multiplier morphisms is separately continuous for the strict topology we have f e i = (cid:80) j ∈ I f j e ∗ j e i = f i for every i in I . If f (cid:48) = ( L (cid:48) , R (cid:48) ) is a second multiplier morphism from C to D such that f (cid:48) e i = f i , then f = f (cid:88) i e i e ∗ i = (cid:88) i f e i e ∗ i = (cid:88) i f (cid:48) e i e ∗ i = f (cid:48) (cid:88) i e i e ∗ i = f (cid:48) . emark 5.9. An important distinction between our notion of orthogonal sums andorthogonal AV-sums is that orthogonal sums in our sense are always unital objects,whereas orthogonal AV-sums can only be unital objects if the family I is finite. In fact, ifthe AV -sum is unital, then Lemma 5.6 implies the norm convergence of the sum (cid:80) i ∈ I e i e ∗ i .Due to Remark 5.9 the only reasonable way of comparing the two notions of orthogonalsums is to compare the image of an orthogonal AV-sums under the functor C → M C fromLemma 5.4 with the corresponding orthogonal sum in M C . Note that Ob( C ) = Ob(M C ).So let C be in C ∗ Cat nu , and let ( C i ) i ∈ I be a family of objects of C . Let furthermore( C, ( e i ) i ∈ I ) be an orthogonal AV-sum. Proposition 5.10. ( C, ( e i ) i ∈ I ) is an orthogonal sum in the C ∗ -category M C in the senseof Definition 3.16.Proof. We have to show that the associated multiplier morphisms (3.9) and (3.10) arebijective for every object D of C . It suffices to discuss the case of right multipliers. Theother case follows from applying the involution.Before showing that (3.9) is bijective, we show two preparatory statements.1. If E is an object of C , then K ( C, E ) is strictly dense in
MHom C ( C, E ).2. For any right multiplier morphism R in RM( D, C ) (see Definition 3.11) and object E of C the map R E : K ( C, E ) R → Hom C ( D, E ) → MHom C ( D, E )is right-strictly continuous. We start with Assertion 1. So let M = ( L, R ) be in
MHom C ( C, E ). We have the chain ofequalities M = M id C ! = M (cid:88) i ∈ I e i e ∗ i ! = (cid:88) i ∈ I M e i e ∗ i . For one marked with ! we used our assumption to the effect that the sum (cid:80) i ∈ I e i e ∗ i converges strictly to the identity multiplier morphism of C , and for the one marked with !!we use that composition M ◦ − is strictly continuous. Recall that K ( C, E ) is generated by(multiplier) morphisms of the form N i e ∗ i for (multiplier) morphisms N i : C i → E . Henceeach summand of the sum (cid:80) i ∈ I M e i e ∗ i belongs to K ( C, E ). Therefore K ( C, E ) is strictlydense in
MHom C ( C, E ). That is to say, continuous if we equip its domain and target with the norms (5.6). We do not know if R E is in general also left-strictly continuous.
39e now show the Assertion 2. Let R be in RM( D, C ). Let ( f i ) i ∈ I be a net in K ( C, E )converging right-strictly in
MHom C ( C, E ) to a multiplier morphism f = ( L, R ). We have toshow that ( R E ( f i )) i ∈ I converges right-strictly in MHom C ( D, E ). We write ( R E ( f i )) i ∈ I =( L R i , R R i ). Let H be an object of C and h be in Hom C ( E, H ). We must show that ( R R i ( h )) i ∈ I converges in norm.Since ( f i ) i ∈ I converges right-strictly to f , ( hf i ) i ∈ I converges in norm to R H ( h ). Since hf i belongs to K ( C, H ) for every i in I and K ( C, H ) is norm-closed also R ( h ) is in K ( C, H ).It follows that ( R H ( hf i )) i ∈ I converges in norm to R H ( R ( h )). We have R H ( hf i ) = R E ( f i )( h ) = R R i ( h ) . Hence ( R R i ( h )) i ∈ I converges in norm to R H ( R ( h )) finishing the verification of Assertion 2.We now come back to our original problem. We first show that m RD in (3.9) is injective.Let f be a non-zero morphism in MHom C ( D, C ). We have to show that the correspondingright multiplier R ( f ) := m RD ( f ) in RM( D, C ) is non-zero, too. By Assertions 1 and 2 themap R ( f ) C : K ( C, C ) → MHom C ( D, C ) has a well-defined and unique (since the right-stricttopology is Hausdorff) right-strictly continuous extension to a map (cid:101) R ( f ) C : MHom C ( C, C ) → MHom C ( D, C ). By uniqueness of the extension this map is of course still given by − ◦ f .We have (cid:101) R ( f ) C ( id C ) = id C ◦ f = f . This implies that 0 (cid:54) = (cid:101) R ( f ) C and hence R ( f ) (cid:54) = 0again by the uniqueness assertion.To show surjectivity of the transformation m RD in (3.9) we let R be in RM( D, C ). Weconsider the right-strictly continuous extension (cid:101) R C of R C : K ( C, C ) → MHom C ( D, C ) to
MHom C ( C, C ), and apply it to the identity multiplier morphism id C to get the multipliermorphism f := (cid:101) R C ( id C ) in MHom C ( D, C ). We claim that m RD ( f ) = R . To show this, let E be any object of C and consider R E : K ( C, E ) → MHom C ( D, E ). For ϕ in K ( C, E ) we have R E ( ϕ ) = R E ( ϕ id C ) = (cid:101) R E ( ϕ id C ) ! = ϕ (cid:101) R C ( id C ) = ϕf , where in the marked equality we have used that the right-strictly continuous extension (cid:101) R of R still satisfies the algebraic conditions of a natural transformation of C -vectorspace valued functors MHom C ( C, − ) → MHom C ( D, − ). This proves that R E is given as rightmultiplication by f .The following is a partial converse of Proposition 5.10. Let C be in C ∗ Cat , and let( C i ) i ∈ I be a family of objects of C . We furthermore consider an object C and a mutuallyorthogonal family ( e i ) i ∈ I of isometries e i : C i → C . The left strict topology relative to( e i ) i ∈ I on End C ( C ) is generated by the seminorms f (cid:55)→ (cid:107) f k (cid:107) for all k in K ( D, C ) forsome object D of C . Similarly, the right strict topology relative to ( e i ) i ∈ I on End C ( C ) isgenerated by the seminorms f (cid:55)→ (cid:107) hf (cid:107) for all h in K ( C, D ) for some object D of C . Proposition 5.11. . The sum (cid:80) i ∈ I e i e ∗ i converges to id C left and right strictly relative to ( e i ) i ∈ I .2. If ( C, ( e i ) i ∈ I ) represents the orthogonal sum of ( C i ) i ∈ I in the sense of Definition 3.16,then the left strict and the right strict topologies relative to ( e i ) i ∈ I on End C ( C ) areHausdorff.Proof. Recall that K ( D, C ) is generated by morphisms f : D → C of the form f = e i ˜ f formorphisms ˜ f : D → C i . It follows immediately that (cid:80) i ∈ I e i e ∗ i f converges in norm to f .The case of the right strict convergence relative to ( e i ) i ∈ I is analogous.For the second assertion (the case of the left strict topology) we have to show that forany non-zero f in Hom C ( C, C ) there exists an object E of C and a morphism g in K ( E, C )such that f g is non-zero. Since f is non-zero, it follows from Corollary 4.2.1 that thereexists j in I such that f e j is non-zero. This does the job since e j is in K ( C j , C ). Remark 5.12.
If the family ( e i ) i ∈ I has infinitely many non-zero members, then the sum (cid:80) i ∈ I e i e ∗ i can not converge in the strict topology, see Remark 5.9. In this case the leftand the right strict topologies on End C ( C ) are strictly weaker than the strict topology. Inparticular, ( C, ( e i ) i ∈ I ) can not be an orthogonal AV-sum. Remark 5.13.
In the proof of Proposition 3.22 we have exhibited another topology on A := Hom C ( C, C ) with respect to which (cid:80) i ∈ I e i e ∗ i converges to the identity of C . Let A ∗∗ denote the double commutant of the image of A in its universal representation. Then (cid:80) i ∈ I e i e ∗ i converges in the strong operator topology on A ∗∗ to the identity 1 A = id C . C ∗ -categories and orthogonalsums The property of being an orthogonal sum of a given family of objects in general dependson the surrounding category. In Lemma 4.4 we have seen that if all relevant objects belongto a full subcategory, then a sum in the larger category is also one in the smaller. In thissection we consider the reverse situation and ask to what extend an orthonormal sum inthe smaller category is also one in the larger. In particular we study the case when theembedding is not full. Due to Remark 3.18 this discussion is only interesting for infiniteorthogonal sums.Let C be in C ∗ Cat and assume that D is a closed unital sub- C ∗ -category of C . Let ( C i ) i ∈ I be a family of objects of D and assume that it admits an orthogonal sum ( D, ( e (cid:48) i ) i ∈ I ) in D and an orthogonal sum ( C, ( e i ) i ∈ I ) in C .41 roposition 6.1.
1. There exists a unique isometry h : C → D in C such that he i = e (cid:48) i for all i in I .2. For any object E of D the maps Hom D ( D, E ) → Hom C ( C, E ) , f (cid:55)→ f h (6.1) and Hom D ( E, D ) → Hom C ( E, C ) , f (cid:55)→ h ∗ f (6.2) are isometric inclusions.3. The map End D ( D ) → End C ( C ) , f (cid:55)→ h ∗ f h (6.3) identifies the C ∗ -algebra End D ( D ) with a corner of End C ( C ) . Here we omit to write the inclusion map of D to C . Note that h identifies C with asubobject of D considered as an object of C . Proof.
We start with Assertion 1. We apply the Corollary 4.1.2 to the family ( h (cid:48) i ) i ∈ I ofmorphisms h (cid:48) i := e (cid:48) i : C i → D to get a unique morphism h : C → D satisfying he i = e (cid:48) i forall i in I .The composition h ∗ h : C → C is a morphism in C which satisfies e ∗ j h ∗ he i = ( e (cid:48) j ) ∗ e (cid:48) i = e ∗ j e i : C i → C j for all i, j in I . By Corollary 4.2.3 these equalities together imply that h ∗ h = id C , i.e.,that h is an isometry. Then p := hh ∗ (6.4)is a projection in End C ( D ).We now show Assertion 2. Let E be an object of D . Then we consider the diagram Hom D ( D, E ) (cid:15) (cid:15) ! (cid:42) (cid:42) f (cid:55)→ fh (cid:47) (cid:47) Hom C ( C, E ) Hom C ( D, E ) f (cid:55)→ fp (cid:47) (cid:47) Hom C ( D, E ) p g (cid:55)→ gh (cid:79) (cid:79) We must show that the upper horizontal map is isometric. We first observe that the rightvertical map is an isometry with inverse l (cid:55)→ lh ∗ . Note that lh ∗ belongs to Hom C ( D, E ) p since lh ∗ = l ( h ∗ h ) h ∗ = ( lh ∗ ) p . It remains to show that the map marked by ! is isometric.42he argument is as follows. In order to distinguish left-multipliers formed in D or in C we use the notationLM D ( D, E ) :=
Hom bd Fun ( D op , Ban ) ( K D ( − , D ) , Hom D ( − , E )) , LM C ( C, E ) :=
Hom bd Fun ( C op , Ban ) ( K C ( − , C ) , Hom C ( − , E )) . We define LM C | D ( D, E ) :=
Hom bd Fun ( D op , Ban ) ( K D ( − , D ) , Hom C ( − , E ))and get a restriction map −| D : LM C ( D, E ) → LM C | D ( D, E ) . Note that we also have a canonical inclusionLM D ( D, E ) → LM C | D ( D, E ) (6.5)given by the inclusion of
Hom D ( − , E ) into Hom C ( − , E ). We consider the mapLM C ( C, E ) → LM C ( D, E ) , L (cid:55)→ Lh , (6.6)where Lh ∗ is defined by Lh ∗ ( k ) := L ( h ∗ k ) for every k in K C ( F, D ). It is well-defined sinceby Assertion 1 we have h ∗ e (cid:48) , ∗ i = e i for all i in I which implies that left-composition with h ∗ gives a well-defined natural transformation h ∗ − : K C ( − , D ) → K C ( − , C ). The map (6.6)is an isomorphism with inverse given by mapping L (cid:48) in LM C ( D, E ) to L (cid:48) h defined similarlyas above. In fact ( Lh ∗ ) h ( k ) = L ( hh ∗ k ) = L ( k ) for all k in K C ( F, D ). Furthermore, for L (cid:48) in LM C ( D, E ) we have ( L (cid:48) h ∗ ) h ( k (cid:48) ) = L (cid:48) ( hh ∗ k (cid:48) ) = L (cid:48) ( k (cid:48) ) for all k (cid:48) in K C ( F, D ) since byAssertion 1 we have hh ∗ e (cid:48) i = e (cid:48) i for all i in I and therefore p = hh ∗ acts as the identity on K C ( F, D ).We consider the diagram
Hom C ( C, E ) (3.10) ,m LE ∼ = (cid:47) (cid:47) l (cid:55)→ lh ∗ ∼ = (cid:15) (cid:15) LM C ( C, E ) (6.6) ∼ = (cid:15) (cid:15) Hom C ( D, E ) p incl (cid:47) (cid:47) Hom C ( D, E ) (3.10) ,m LD (cid:47) (cid:47) LM C ( D, E ) −| D (cid:15) (cid:15) LM C | D ( D, E ) Hom D ( D, E ) ! (cid:79) (cid:79) (3.10) ∼ = (cid:47) (cid:47) LM D ( D, E ) (6.5) !! (cid:79) (cid:79) We want to show that the map marked by ! (given by f (cid:55)→ f p ) is an isometry. Multiplyingfrom the left by p on K D ( − , D ) is well-defined and acts as the identity since pe (cid:48) i = e (cid:48) i forall i as observed above. Therefore the map marked by !! is equal to the canonical inclusion(6.5). 43n order to prove that the map marked by ! is isometric, we first note that all maps in theabove diagram are non-expansive. Furthermore, the associated left multiplier map (3.10)is isometric by Lemma 3.22, and the canonical inclusion (6.5) (i.e., the map marked !!) isisometric since the inclusion of Hom D ( − , E ) into Hom C ( − , E ) is isometric since D is closedin C . The combination of these facts implies that ! is isometric.The other assertions of the lemma are shown by similar arguments.For the next proposition we retain the notation introduced before Proposition 6.1. Proposition 6.2.
1. If
Hom D ( C i , E ) = Hom C ( C i , E ) for every i in I , then (6.1) is an isomorphism.2. If Hom D ( E, C i ) = Hom C ( E, C i ) for every i in I , then (6.2) is an isomorphism.3. If Hom D ( C i , C j ) = Hom C ( C i , C j ) for every i, j in I , then (6.3) is an isomorphism.Proof. We show Assertion 1. We have already seen in Proposition 6.1.1 that (6.1) is anisometric inclusion. It therefore suffices to show that this map is also surjective.Let g be in Hom C ( C, E ). Then by assumption ge i : C i → E is a morphism in D for every i in I . We apply Corollary 4.1.2 (to the orthogonal sum D in D ) to the family of morphisms( ge i ) i ∈ I in order to get a morphism f (cid:48) : D → E in D with f (cid:48) e (cid:48) i = ge i for every i in I . ByProposition 6.1.1 the composition f (cid:48) h satisfies f (cid:48) he i = f (cid:48) e (cid:48) i for every i in I , and henceCorollary 4.2.1 implies that f (cid:48) h = g . Therefore f (cid:48) is a preimage of g under (6.1).The Assertion 2 follows from Assertion 1 by using the involution, and the argument forAssertion 3 is similar.We still retain the notation introduced before Proposition 6.1. Proposition 6.3. If End D ( D ) = End C ( D ) , then the morphism h : C → D constructedin the Proposition 6.1.1 is an isomorphism between the orthogonal sums ( C, ( e i ) i ∈ I ) and ( D, ( e (cid:48) i ) i ∈ I ) .Proof. It suffices to show that hh ∗ = id D in End C ( D ). By assumption h ∗ belongs to End D ( D ). Hence we may apply Corollary 4.2.3 to the sum ( D, ( e (cid:48) i ) i ∈ I ) in the category D and the identities e (cid:48) , ∗ i hh ∗ e (cid:48) j = e ∗ i e j for all i, j in I in order to conclude that hh ∗ = id D .44 xample 6.4. In this example we construct a situation where the inclusion
End D ( D ) ⊆ End C ( D ) is proper and h is not an isomorphism. This shows that the assumption in theProposition 6.3 can not be dropped.Let X be a countably infinite set. We let ∼ be the equivalence relation on the powerset P ( X ) of X given by A ∼ B if and only if the symmetric difference A ∆ B is finite.Let [ − ] : P ( X ) → P ( X ) / ∼ be the quotient map. The set P ( X ) is a Boolean algebraunder the operations of forming unions, intersections and complements. These operationsdescend to the quotient P ( X ) / ∼ . Using Stone’s representation theorem for Booleanalgebras [Sto36], we get a set Y and an injective homomorphism of Boolean algebras s : ( P ( X ) / ∼ ) → P ( Y ).For every x in X let p x be the orthogonal projection in B ( (cid:96) ( X )) onto the one-dimensionalsubspace spanned by x , and for a subset A of X we consider the orthogonal projection p A := (cid:80) x ∈ A p x in B ( (cid:96) ( X )) (the sum is strongly convergent). Analogously we define forevery subset B of Y an orthogonal projection q B in B ( (cid:96) ( Y )).We will use the notation conventions as in Example 4.10 in order to denote subspaces ofthe algebra B ( (cid:96) ( Y ∪ X )). We define a C ∗ -category C as follows:1. objects: The set of objects of C is X ∪ { X, X } .2. morphisms: The morphisms of C are given as subspaces of B ( (cid:96) ( Y ∪ X )) as follows:a) End C ( x ) := B ( (cid:96) ( { x } )) for x in X .b) End C ( X ) is the subalgebra of B ( (cid:96) ( Y ∪ X )) generated by the operators p (cid:48) A + q s ( A ) for all subsets A of X and B ( (cid:96) ( X )), where p (cid:48) x is p x considered as an elementof End C ( X ).c) End C ( X ) := B ( (cid:96) ( X )).d) Hom C ( x, x (cid:48) ) := B ( (cid:96) ( { x } ) , (cid:96) ( { x (cid:48) } )).e) Hom C ( x, X ) := B ( (cid:96) ( { x } ) , (cid:96) ( X )) and Hom C ( X, x ) := B ( (cid:96) ( X ) , (cid:96) ( { x } )).f) Hom C ( x, X ) := B ( (cid:96) ( { x } ) , (cid:96) ( X )) and Hom C ( X , x ) := B ( (cid:96) ( X ) , (cid:96) ( { x } )).g) Hom C ( X, X ) := B ( (cid:96) ( X )) and Hom C ( X , X ) := B ( (cid:96) ( X )).3. involution and composition: are induced from B ( (cid:96) ( Y ∪ X )).We let e x in B ( (cid:96) ( { x } ) , (cid:96) ( X )) be the canonical inclusion of (cid:96) ( { x } ) into (cid:96) ( X ). We write e (cid:48)(cid:48) x for the corresponding morphism from x to X under the identification 2f. The pair( { X } , ( e (cid:48)(cid:48) x ) x ∈ X ) is an orthogonal sum of the family of objects ( x ) x ∈ X in C . This follows45rom a combination of Example 4.11 and Proposition 6.3 applied to the full subcategory ofon the objects X ∪ { X } of C . We now describe an isometric embedding of the category X from Example 4.10 onto a subcategory D of C .1. objects: The embedding sends the objects x and { X } of X to the correspondingobjects of C with the same name.2. morphisms:a) The map End X ( x ) → End C ( x ) is given by the identity of B ( (cid:96) ( { x } )).b) The map End X ( X ) → End C ( X ) sends the generator p A to p (cid:48) A + q s ( A ) .c) Then map Hom X ( x, X ) → Hom C ( x, X ) is the canonical inclusion C e x → B ( (cid:96) ( { x } ) , (cid:96) ( X )) . d) The map Hom X ( X, x ) → Hom C ( X, x ) is the canonical inclusion C e ∗ x → B ( (cid:96) ( X ) , (cid:96) ( { x } )) . The image in D of the morphism e x in X will be denoted by e (cid:48) x . In order to see that this iscompatible with the composition note that e x e ∗ x = p x in X is send to p (cid:48) x + q s ( { x } ) = p (cid:48) x = e (cid:48) x e (cid:48) , ∗ x in C since s ( { x } ) = ∅ since { x } (cid:39) ∅ .It is easy to see that End X ( X ) → End C ( X ) is injective and hence isometric. We concludethat D is an isometric copy of X in C . Note that End X ( X ) → End C ( X ) is not surjective.By Example 4.10 the pair ( X, ( e (cid:48) x ) x ∈ X ) is an orthogonal sum of the family ( x ) x ∈ X in D .The morphism h : X → X constructed in Lemma 1 is given by the identity of B ( (cid:96) ( X ))under the identification 2g. The projection hh ∗ in End C ( X ) is given by the image of1 B ( (cid:96) ( X )) in End C ( X ) under the identification 2b. It is not the identity in End C ( X ) since,e.g., hh ∗ ( p (cid:48) X + q s ( X ) ) = p (cid:48) X (cid:54) = p (cid:48) X + q s ( X ) . We conclude that ( X, ( e (cid:48) x ) x ∈ X ) does not representthe orthogonal sum of ( x ) x ∈ X in C anymore. Example 6.5.
We retain the notation from Example 6.4. Let E be the full subcategory of C with the same objects X ∪ { X } as D . This C ∗ -category does not have any orthogonalsum anymore. α -additive C ∗ -categories Let α be a very small cardinal. Let C be in C ∗ Cat .46 efinition 7.1. C is said to be α -additive (less than α -additive) if it admits orthogonalsums for all families indexed by sets of cardinality α (less than α ). Example 7.2. If C is less than ω -additive, then it admits all finite orthogonal sums.Let F : C → D be a morphism in C ∗ Cat . Definition 7.3. F is α -additive if for every family ( C i ) i ∈ I of objects in C indexed by setsof cardinality at most α and every orthogonal sum ( C, ( e i ) i ∈ I ) in C of this family the pair ( F ( C ) , ( F ( e i )) i ∈ I ) is an orthogonal sum in D of the family ( F ( C i )) i ∈ I . We let C ∗ Cat α add denote the full subcategory of C ∗ Cat of α -additive C ∗ -categories. Wefurther denote by (cid:92) C ∗ Cat α add the wide subcategory of C ∗ Cat α add of α -additive C ∗ -functors.By Corollary 6.3 the category (cid:92) C ∗ Cat α add in particular contains all full inclusions betweenobjects of C ∗ Cat α add .Our next lemma shows that sums in a product can be detected factorwise. We consider afamily ( C j ) j ∈ J in C ∗ Cat . Let further ( C i ) i ∈ I be a family of objects in (cid:81) j ∈ J C j . Then forevery i in I we have C i = ( C i,j ) j ∈ J for a family of objects C i,j in C j . For every j in J weconsider a pair ( S j , ( e j,i ) i ∈ I ) where ( e j,i ) i ∈ I is a mutually orthogonal family of morphisms e j,i : C i,j → S j Lemma 7.4.
The following assertions are equivalent.1. The pair ( S j , ( e j,i ) i ∈ I ) is an orthogonal sum of the family ( C i,j ) i ∈ I for every j in J .2. The pair (( S j ) j ∈ J , (( e j,i ) j ∈ J ) i ∈ I ) is a sum of the family ( C i ) i ∈ I in (cid:81) j ∈ J C j .Proof. We first show using Proposition 4.3 that Assertion 1 implies Assertion 2. We verifyAssumption 4.3.1. The verification of 4.3.2 is similar.Assume that ( h i ) i ∈ I is a square summable family of morphisms h i : ( D j ) j ∈ J → C i in (cid:81) j ∈ J C j . Then h i = ( h i,j ) j ∈ J for certain morphisms h i,j : D j → C i,j . Square integrabilitymeans that C := sup j ∈ J sup K (cid:13)(cid:13) (cid:88) k ∈ K h ∗ k,j h k,j (cid:13)(cid:13) ≤ ∞ , where K runs over all finite subsets of I . In particular, the family ( h i,j ) i ∈ I is squareintegrable for every j in J . Since by assumption ( S j , ( e j,i ) i ∈ I ) is an orthogonal sum of the47amily ( C i,j ) i ∈ I in C j for every j in J we get a unique morphism f j : D j → S j such that f j e j,i = h i,j for every i in I . Furthermore, (cid:107) f j (cid:107) ≤ C . It follows that ( f j ) j ∈ J : ( D j ) j ∈ J → ( S j ) j ∈ J is the unique morphism in (cid:81) j ∈ J C j such that ( f j ) j ∈ J ( e j,i ) j ∈ J = h i for every i in I .We conclude Assertion 2.We now show that Assertion 2 implies Assertion 1. We fix j in J and show by usingProposition 4.3 that ( S j , ( e j ,i ) i ∈ I ) is an orthogonal sum of the family ( C i,j ) i ∈ I . We againonly verify Assumption 4.3.1 since the case of 4.3.2 is similar.Let D j be an object of C j and consider a square integrable family ( h i,j ) i ∈ I of morphisms D j → C i,j in C j . We extend D j to an object ( D j ) j ∈ J in (cid:81) j ∈ J C j . We can, e.g., take D j := S j for all j (cid:54) = j . Furthermore, for every i in I we extend h i,j by zero to a family h i := ( h i,j ) j ∈ J . Then ( h i ) i ∈ I is square integrable. Since (( S j ) j ∈ J , (( e j,i ) j ∈ J ) i ∈ I ) is a sumof the family ( C i ) i ∈ I in (cid:81) j ∈ J C j we can find a unique morphism f = ( f j ) j ∈ J : ( D j ) j ∈ J → ( S j ) j ∈ J with ( f j ) j ∈ J ( e j,i ) j ∈ J = h i for every i in I . Then f j : D j → S j is a morphism suchthat f j e j ,i = h i,j for every i in I . We must check that it is uniquely determined by thiscondition. Indeed, every other choice f (cid:48) j satisfying this condition can be extended byzero to a family ( f (cid:48) j ) j ∈ J satisfying ( f (cid:48) j ) j ∈ J ( e j,i ) j ∈ J = h i for every i in I . By the uniquenessstatement above we conclude that f j = f (cid:48) j . Corollary 7.5.
The categories C ∗ Cat α add and (cid:92) C ∗ Cat α add admit all very small products.Proof. The case of C ∗ Cat α add is an immediate consequence of Lemma 7.4.In order to show that (cid:92) C ∗ Cat α add admits products we show that the structure maps ofthe product in C ∗ Cat α add preserve products and present the product in C ∗ Cat α add as aproduct in (cid:92) C ∗ Cat α add . First of all, the Lemma 7.4 implies that the projections onto thefactors of a product preserve orthogonal sums. Let C be in C ∗ Cat α add and let ( F k ) k ∈ J bea family of morphisms F k : C → D k in (cid:92) C ∗ Cat α add . Then it again follows from Lemma 7.4that ( F k ) k ∈ J : C → (cid:81) j ∈ J D j preserves orthogonal sums.The category C ∗ Cat has the structure of a 2-category whose 2-morphisms are unitarynatural transformations between morphisms. Let us recall the construction of the categoryof 2-equalizers in C ∗ Cat . We consider a diagram C F (cid:38) (cid:38) G (cid:57) (cid:57) D . Then (a representative up to unitary equivalence) of the category Eq ( F, G ) of 2-equalizersbetween F and G in C ∗ Cat is given as follows:48. objects: An object of Eq ( F, G ) is a pair (
C, u ) of an object C in C and a unitaryisomorphism u : F ( C ) → G ( C ) in D .2. morphisms: A morphism ( C, u ) → ( C (cid:48) , u (cid:48) ) in Eq ( F, G ) is a morphism f : C → C (cid:48) in C such that u (cid:48) ◦ F ( f ) = G ( f ) ◦ u .3. involution and composition: These structures are inherited from C . Lemma 7.6.
The category (cid:92) C ∗ Cat α add is closed under forming -equalizers in C ∗ Cat .Proof.
Let (( C i , u i )) i ∈ I be a family of objects in Eq ( F, G ). Then we can choose a sum( C, ( e i ) i ∈ I ) of the family ( C i ) i ∈ I in C . Since the functors F and G preserve sums the pair( F ( C ) , ( F ( e i )) i ∈ I ) is a sum of the family ( F ( C i )) i ∈ I , and the pair ( G ( C ) , ( G ( e i )) i ∈ I ) is asum of the family ( G ( C i )) i ∈ I . The family of unitaries ( u i ) i ∈ I is a unitary isomorphismbetween ( F ( C i )) i ∈ I and ( G ( C i )) i ∈ I . By the uniqueness of orthogonal sums up to uniqueunitary isomorphism (Lemma 3.19) there is a unique unitary u : F ( C ) → G ( C ) such that uF ( e i ) = G ( e i ) u i (7.1)for every i in I . The pair ( C, u ) is then an object in Eq ( F, G ). We claim that (
C, u )represents the sum of the family (( C i , u i )) i ∈ I in Eq ( F, G ). First of all the relations (7.1)show that e i : ( C i , u i ) → ( C, u ) is a morphism in Eq ( F, G ) for every i in I . We now applyProposition 4.3 and verify its Assumption 4.3.1. The verification of 4.3.2 is similar. Let( D, v ) be an object of Eq ( F, G ) and let ( h i ) i ∈ I be a square summable family of morphisms h i : ( D, v ) → ( C i , u i ). Then ( h i ) i ∈ I is square summable as a family of morphisms in C .Hence there exists a unique morphism h : D → C in C such that e ∗ i h = h i for all i in I .We have G ( e ∗ i ) G ( h ) v = G ( h i ) v = u i F ( h i ) = u i F ( e ∗ i ) F ( h ) (7.1) = G ( e ∗ i ) uF ( h )for every i in I . This implies that G ( h ) v = uF ( h ). Hence h is a morphism ( D, v ) → ( C, u )in Eq ( F, G ). Remark 7.7.
Since sums are unique up to unitary isomorphisms (as opposed to equality)we can not expect a version of Lemma 7.6 where Eq is replaced by the equalizer in theordinary category C ∗ Cat . Corollary 7.8.
The category (cid:92) C ∗ Cat α add admits all very small -categorical limits.Proof. The existence 2-categorical limits follows from the existence of products and two-categorical equalizers. We use Lemma 7.5 and 7.6.49 emma 7.9.
For every ordinal κ with κ > α the category (cid:92) C ∗ Cat α add admits colimits for κ -filtered diagrams with isometric structure morphisms.Proof. The proof is an exercise based on Proposition 4.3.
We fix a very small regular cardinal α . If C is in C ∗ Cat , then one could try to constructa functorial < α -additive completion of C . Such constructions have been considered for α = ω (completion w.r.t. finite sums) in [DL98, Del12], and for AV-sums in [AV]. In viewof the Corollaries 4.7 and 4.9 we can interpret the closure of the essential image of thefunctor M : C → Hilb ( A ( C )) from (4.6) under all sums of families of cardinality < α asan < α -additive completion of C .In the case of finite sums, if one works in the 2- or ∞ -category C ∗ Cat ∞ , i.e., afterlocalization at the unitary equivalences, then these constructions have the expecteduniversal property, see Remark 8.6 for a precise statement. The goal of the present sectionis to show that completion functors w.r.t. infinite sums can not have similar good universalproperties. The precise formulation is given in the Proposition 8.1.Recall that C ∗ Cat α add ( C ∗ Cat <α add ) denotes the full subcategory of C ∗ Cat of α -additive(less than α -additive) C ∗ -categories (Definition 7.1).We let (cid:96) : C ∗ Cat → C ∗ Cat ∞ denote the Dwyer–Kan localization of C ∗ Cat at the set of unitary equivalences in the realmof ∞ -categories. This ∞ -category can be modelled by a combinatorial model categorystructure on C ∗ Cat . In particular, C ∗ Cat ∞ is a presentable ∞ -category. For details werefer to [Bun19]. For a very small cardinal α we let C ∗ Cat α add ∞ ( C ∗ Cat <α add ∞ ) denote thefull subcategories of C ∗ Cat ∞ of α -additive (less than α -additive) C ∗ -categories. Proposition 8.1.
1. If α ≥ ω , then C ∗ Cat α add is not a set of fibrant objects of a Bousfield localization ofthe combinatorial model category C ∗ Cat . Furthermore, there is no adjunction L : C ∗ Cat ∞ (cid:28) C ∗ Cat α add ∞ : incl .
2. If α > ω , then C ∗ Cat <α add is not a set of fibrant objects of a Bousfield localizationof the combinatorial model category C ∗ Cat . Furthermore, there is no adjunction L : C ∗ Cat ∞ (cid:28) C ∗ Cat <α add ∞ : incl . ω -additive categories (see theRemark 8.6) showing that the assumption α > ω in Proposition 8.1.2 is necessary.We first construct in Example 8.2 an isometric inclusion of C ∗ -categories which does notpreserve orthogonal sums. Similar situations already occured in the Examples 6.4 & 6.5,but this one will be the main ingredient in our argument proving the assertions above. Example 8.2.
Let C be in C ∗ Cat . The object (cid:81) N C in C ∗ Cat contains the ideal I ofall morphisms ( f k ) k ∈ N such that lim k →∞ (cid:107) f k (cid:107) = 0. We let D := ( (cid:81) N C ) / I and considerthe functor h : C diag −−→ (cid:89) N C pr −→ D . (8.1)We note that h is an isometric inclusion.We consider a family of non-zero objects ( C i ) i ∈ N in C which admits an orthogonal sum( C, ( e i ) i ∈ N ). We claim that then ( h ( C ) , h ( e i ) i ∈ N ) in D does not represent the sum of thefamily ( h ( C i )) i ∈ N .To show the claim we consider the morphism f := ( (cid:80) i>k e i e ∗ i ) k ∈ N in End (cid:81) N C ( diag ( C ))and let ˜ f := pr ( f ) (8.2)in End D ( h ( C )) be the image of f in D . We note that (cid:107) ˜ f (cid:107) = 1. We further observethat for every i in I we have f ◦ diag ( e i ) ∈ I , and consequently ˜ f ◦ h ( e i ) = 0. Thisimplies that the image of ˜ f in LM(( h ( C ) , h ( e i ) i ∈ N ) , h ( C )) vanishes. Therefore the map(3.10), i.e., Hom D ( h ( C ) , h ( C )) → LM(( h ( C ) , h ( e i ) i ∈ N ) , h ( C )) is not injective, and therefore( h ( C ) , h ( e i ) i ∈ N ) does not represent the sum of the family ( h ( C i )) i ∈ N .Let I be a very small filtered category and C : I → C ∗ Cat be a functor. Since C ∗ Cat iscocomplete, the colimit colim I C exists. For every i in I we have the canonical functor ι i : C ( i ) → colim I C . Lemma 8.3.
Assume that the structure maps of the diagram C are isometric. Then forevery i in I the canonical functor ι i : C ( i ) → colim I C is isometric.Proof. We let F : C ∗ Cat → ∗ Cat C be the inclusion of C ∗ Cat as a full subcategory of thecategory ∗ Cat C of C -linear ∗ -categories. Then colim I F ( C ) has the following description.1. objects: The objects of the colimit of C -linear ∗ -categories is the colimit of the setof objects, i.e., Ob( colim I F ( C )) ∼ = colim I Ob( F ( C )). For every i in I and C inOb( F ( C ( i ))) we let ι i ( C ) denote the object in Ob( colim I F ( C )) represented by C .51. morphisms: For every two objects ¯ C and ¯ C (cid:48) in colim I F ( C ) we can (because I isfiltered) find i in I and objects C and C (cid:48) in C ( i ) such that ¯ C = ι i ( C ) and ¯ C (cid:48) = ι i ( C (cid:48) ).We then have Hom colim I F ( C ) ( ¯ C, ¯ C (cid:48) ) := colim ( i → i (cid:48) ) ∈ I i/ Hom C ( i (cid:48) ) ( C ( i → i (cid:48) )( C ) , C ( i → i (cid:48) )( C (cid:48) )) , (8.3)where I i/ denotes the slice category of objects under i in I .3. composition and involution: They are defined in the canonical manner.We now define a norm on colim I F ( C ) as follows. If ¯ f : ¯ C → ¯ C (cid:48) is any morphism in colim I F ( C ), then there exists ( i → i (cid:48) ) ∈ I i/ and a morphism f in C ( i (cid:48) ) which is send to¯ f under the canonical map associated to the colimit in (8.3). We then define (cid:107) ¯ f (cid:107) := (cid:107) f (cid:107) .Since the structure maps C ( i ) → C ( i (cid:48) ) for morphisms i → i (cid:48) are all isometric this valuedoes not depend on the choices. One furthermore checks that the norm on colim I F ( C ) issub-multiplicative and satisfies the C ∗ -identity and the C ∗ -inequality. These propertiesare all inherited from corresponding properties of the stages of the colimit and the factthat the structure maps are isometric.We form the completion D := colim I F ( C ) with respect to the norm. This amounts toforming the completion of the morphism spaces and extending the composition and theinvolution by continuity. Then D is an object in C ∗ Cat . The structure maps for thecolimit provide the first map of the composition F ( C ) → colim I F ( C ) → F ( D ) (8.4)in Fun ( I , ∗ Cat C ), where − is the constant I -diagram on − . The second morphism isinduced by the inclusion of the colimit into its completion. Since C ∗ Cat → ∗ Cat C isfully faithful, the inclusion (8.4) is a morphism C → D in Fun ( I , C ∗ Cat ), and hence byadjunction corresponds a functor σ : colim I C → D . If i is in I and f is a morphism in C ( i ), then using that a functor in C ∗ Cat is norm non-increasing we have the followingchain of inequalities (cid:107) f (cid:107) = (cid:107) σ ( ι i ( f )) (cid:107) ≤ (cid:107) ι i ( f ) (cid:107) ≤ (cid:107) f (cid:107) . It implies the desired equality (cid:107) f (cid:107) = (cid:107) ι i ( f ) (cid:107) .Let α be a very small cardinal. We let C ∗ Cat (cid:48) be one of C ∗ Cat α add or C ∗ Cat <α add .We assume that C ∗ Cat (cid:48) is the set of fibrant objects of a Bousfield localization of thecombinatoral model category structure C ∗ Cat . We let L : C ∗ Cat (cid:28) C ∗ Cat (cid:48) , (cid:96) : id → L (8.5)denote a fibrant replacement.Let C be in C ∗ Cat . Note the our goal is to show that such an adjunction can not exist, see Proposition 8.1. emma 8.4. The functor (cid:96) C : C → L ( C ) in (8.5) is an isometric inclusion.Proof. The category
Hilb ( C ) of very small Hilbert spaces admits all very small orthogonalsums. In particular, it is (less than) α -additive, and consequently, the restriction along (cid:96) C : C → L ( C ) induces a bijection (cid:96) ∗ C : Hom ho C ∗ Cat ( L ( C ) , Hilb ( C )) ∼ = −→ Hom ho C ∗ Cat ( C , Hilb ( C )) . (8.6)Let f be a morphism in C . Using Lemma 2.9 we conclude that (cid:107) f (cid:107) max = sup [ σ ] ∈ Hom ho C ∗ Cat ( C , Hilb ( C )) (cid:107) [ σ ]( f ) (cid:107) = sup [ σ (cid:48) ] ∈ Hom ho C ∗ Cat ( L ( C ) , Hilb ( C )) (cid:107) σ (cid:48) ( (cid:96) C ( f )) (cid:107) = (cid:107) (cid:96) C ( f ) (cid:107) max . Note that the elements [ σ ] in Hom ho C ∗ Cat ( C , Hilb ( C )) are unitary equivalence classes offunctors σ : C → Hilb ( C ), hence the value (cid:107) σ ( f ) (cid:107) does not depend on the choice of therepresentative.We let P : C ∗ Cat → C ∗ Cat be the endofunctor which sends C to the category P ( C ) := (cid:16) (cid:89) N C (cid:17)(cid:46) I , where I is as in Example 8.2. We have a canonical isometric inclusion h C : C → P ( C ).We now construct by transfinite induction a filtered family ( C κ ) κ in C ∗ Cat (cid:48) .Let C be in C ∗ Cat (cid:48) .1. We set C := C .2. If κ is a successor, then we define C κ := L ( P ( C κ − )), where L is as in (8.5). Wehave an isometric inclusion C κ − h C κ − −−−−→ P ( C κ − ) (cid:96) P ( C κ − −−−−−→ C κ , where h C κ − is as in (8.1). In fact, h C κ − is isometric as seen in the Example 8.2,and (cid:96) P ( C κ − ) is isometric by Lemma 8.4.3. If κ is a limit ordinal, then we define C κ := L ( colim β<κ C β ) . β with β < κ we have an isometric inclusion C β i β −→ colim β<κ C β (cid:96) colim β<κ C β −−−−−−−→ C κ , where i β is isometric by Lemma 8.3, and (cid:96) colim β<κ C β is isometric by Lemma 8.4.We now assume that κ is ω -filtered. Lemma 8.5. colim β<κ C β does not admit countable orthogonal sums.Proof. We abbreviate ¯ C := colim β<κ C β . Let ( ¯ C i ) i ∈ N be a family of objects in ¯ C . Assumethat ( ¯ C, (¯ e i ) i ∈ N ) represents an orthogonal sum of this family in ¯ C .Since κ is ω -filtered there exists β with β < κ and representatives ( C i ) i ∈ N and C of theobjects ( ¯ C i ) i ∈ N and ¯ C in C β .Note that colim β<κ C β is the closure of the colimit formed in ∗ Cat C . For every i in N exists a sequence of ordinals ( β i,n ) n ∈ N such that β ≤ β i,n < κ and a sequence ( e i,n ) n ∈ N ofmorphisms e i,n : C i,n → C between the images of C i and C in C β i,n with lim n →∞ ¯ e i,n = ¯ e i in ¯ C . Since κ is ω -filtered there exists β such that β i,n ≤ β < κ for all i and n in N . Sincethe structure maps of the system ( C α ) α are all isometric, the map C β → ¯ C is isometricby Lemma 8.3.We now use the notation C i , C , and ( e i,n ) i,n ∈ N for the images of the corresponding data in C β . Then e i := lim n → N e i,n exists in C β , and ( e i ) i ∈ N is a mutually orthogonal family ofisometries C i → C .Note that C β belongs to C ∗ Cat (cid:48) and therefore admits countable sums. Let ( ˜ C, ˜ e i ) i ∈ N represent the orthogonal sum of the family ( C i ) i ∈ N in C β . Then we can define an isometry u := (cid:88) i ∈ N e i ˜ e ∗ i : ˜ C → C , hence u ∗ u = id ˜ C . We now consider the morphism ˜ f in End P ( C β ) ( h C β ( ˜ C )) as in (8.2) in Ex-ample 8.2. It satisfies (cid:107) ˜ f (cid:107) = 1. Then the morphism ˆ f := h ( u ) ˜ f h ( u ∗ ) in End P ( C β ) ( h C β ( C ))also satisfies (cid:107) ˆ f (cid:107) = 1. We note that u ∗ e i = ˜ e i for every i in N . The relations ˜ f h (˜ e i ) = 0and h (˜ e ∗ i ) ˜ f = 0 imply ˆ f h ( e i ) = h ( u ) ˜ f h ( u ∗ ) h ( e i ) = h ( u ) ˜ f h ( u ∗ e i ) = h ( u ) ˜ f h (˜ e i ) = 0 andsimilarly h ( e i ) ∗ ˆ f = 0 for every i in N .The canonical map P ( C β ) (cid:96) P ( C β ) −−−−→ L P ( C β ) = C β +1 → ¯ C is isometric. Furthermore note that (cid:96) P ( C β ) ◦ h : C β → C β +1 is the structure map of the system ( C α ) α . We conclude that theimage ¯ f of ˆ f in End ¯ C ( ¯ C ) satisfies (cid:107) ¯ f (cid:107) = 1 and ¯ f ¯ e i = 0 and ¯ e ∗ i ¯ f = 0 for every i in N . Asin Example 8.2 this is in conflict with the fact that ( ¯ C, (¯ e i ) i ∈ N ) represents the orthogonalsum of the family ( ¯ C i ) i ∈ N . 54et α be a very small cardinal. Proof of Proposition 8.1.
The argument for the two cases is similar. Let
Cat (cid:48) be either
Cat α add or Cat <α add . It suffices to show that the Bousfield localization on the level ofmodel categories can not exist. Assume by contradiction that it exists. Then the inclusion C ∗ Cat (cid:48) → C ∗ Cat would be accessible. This means that there exists a regular cardinal κ such that C ∗ Cat (cid:48) admits κ -filtered colimits and is generated by κ -compact objects, andthat the inclusion functor preserves κ -filtered colimits. Lemma 8.5 implies that if C ∗ Cat (cid:48) admits κ -filtered colimits, then these are not preserved by the inclusion. Remark 8.6.
Note that a finite additive completion functor L ⊕ : C ∗ Cat → C ∗ Cat <ω hasbeen constructed in [DL98, Sec. 2] and [DT14, Defn. 2.8], and that we have a Bousfieldlocalization C ∗ Cat ∞ (cid:28) C ∗ Cat <ω add ∞ : incl . This shows that the assumption α > ω in Proposition 8.1.2 is necessary.
Remark 8.7.
Let (cid:92) C ∗ Cat α add ∞ be the Dwyer–Kan localization of (cid:92) C ∗ Cat α add at the unitaryequivalences. Note that we have not excluded the existence of an adjunction( < ) α -sum completion : C ∗ Cat ∞ (cid:28) ˆ C ∗ Cat ( < ) α add ∞ : i , where i is induced by the inclusion (cid:92) C ∗ Cat ( < ) α add → C ∗ Cat . In fact, Corollary 7.8 and itsproof is close to saying that i preserves limits. We have not studied this problem further. G -invariantfunctors Assume that C is in C ∗ Cat . Let I be a very small set and assume that C is | I | -additive(see Definition 7.1). Definition 9.1.
We construct a functor (cid:77) I : (cid:89) I C → C (9.1) as follows: . objects: For every object ( C i ) i ∈ I of (cid:81) I C we choose an orthogonal sum ( (cid:77) i ∈ I C i , ( e i ) i ∈ I ) . This determines the action of the functor on objects.2. morphisms: Let ( C i ) i ∈ I and ( C (cid:48) i ) i ∈ I be objects in (cid:81) I C and consider a morphism ( f i ) i ∈ I : ( C i ) i ∈ I → ( C (cid:48) i ) i ∈ I in (cid:81) I C , where f i : C i → C (cid:48) i for all i in I . Then we have sup i ∈ I (cid:107) f i (cid:107) < ∞ and Example 4.14 provides the morphism (cid:77) I ( f i ) i ∈ I := ⊕ i ∈ I f i : (cid:77) i ∈ I C i → (cid:77) i ∈ I C (cid:48) i . By the uniqueness assertion in Example 4.14 this construction is compatible with composi-tions and the involution. Note that the functor (9.1) depends on the choice of the objectsrepresenting the orthogonal sums made in Definition 9.1.1. By Lemma 3.19 a differentchoice here leads to a uniquely unitarily isomorphic functor.Let D and C be in C ∗ Cat , and let ( φ i ) i ∈ I be a family of functors in Hom C ∗ Cat ( D , C ). Weassume that C is | I | -additive and fix a choice for the functor (9.1). Definition 9.2.
We define the orthogonal sum ⊕ i ∈ I φ i : D → C (9.2) of the family ( φ i ) i ∈ I as the composition ⊕ i ∈ I φ i : D diag −−→ (cid:89) i ∈ I D (cid:81) i ∈ I φ i −−−−→ (cid:89) i ∈ I C (cid:76) I −−→ C . Again, this sum depends on the choice adopted for (cid:76) I . A different choice here leads to auniquely unitarily isomorphic functor.Let C be in C ∗ Cat . Definition 9.3. C is flasque if it is additive (see Definition 3.5) and admits an endofunctor S : C → C such that id C ⊕ S is unitarily isomorphic to S . We say that S implements flasqueness of C . Note that the sum id C ⊕ S is defined by(9.2).Let C be in C ∗ Cat . 56 xample 9.4. If C is countably additive, then it is flasque. Indeed, according to theDefinition 9.2 we can construct the endofunctor S := ⊕ N id C : C → C . One easily finds aunitary isomorphism between id C ⊕ S and S .Let G be a group, and let BG be the category with a single object ∗ BG and the monoid ofendomorphisms End BG ( ∗ BG ) := G . If C is any category, then Fun ( BG, C ) is the category of G -objects and equivariant morphisms in C . We have a forgetful functor Fun ( BG, C ) → C which forgets the G -action.For convenience we adopt the following notational convention. We use decorated symbols˜ C, ˜ C (cid:48) to denote G -objects in Fun ( BG, C ), and similarly decorated letters to denotemorphisms ˜ f : ˜ C → ˜ C (cid:48) . In this case we write C := ˜ C ( ∗ BG ) for the underlying object in C ,and f : C → C (cid:48) for the morphism between the underlying objects induced by ˜ f .Let now ˜ C be in Fun ( BG, C ∗ Cat ). In other words, ˜ C is a C ∗ -category with a strict actionof G by automorphisms. Remark 9.5.
Since C ∗ Cat is the underlying category of a (2 , G with relaxed associativityconstraints. The adjective “strict” indicates that here we consider actions in the classicalsense.The (2 , C ∗ Cat also allows to weaken the equivariance condition onmorphisms. This leads to the notion of a weakly invariant functor as defined below.Let ˜ C and ˜ C (cid:48) be in Fun ( BG, C ∗ Cat ). Definition 9.6.
A weakly invariant functor from C to C (cid:48) is a pair ( φ, ρ ) consisting ofthe following data:1. a functor φ : C → C (cid:48) ;2. a family ( ρ = ( ρ ( g )) g ∈ G of unitary isomorphisms ρ ( g ) : φ → g − φg such that for all g, g (cid:48) in G we have g − ρ ( g (cid:48) ) gρ ( g ) = ρ ( g (cid:48) g ) . Remark 9.7.
For ˜ C and ˜ C (cid:48) in Fun ( BG, C ∗ Cat ) we can consider the groupoid of func-tors
Fun C ∗ Cat ( ˜ C , ˜ C (cid:48) ) + and unitary isomorphisms in Fun ( BG,
Grpd ), where G acts byconjugation. The weakly invariant functors are precisely the two-categorial G -invariantsin Fun C ∗ Cat ( ˜ C , ˜ C (cid:48) ) + .If φ : C → C (cid:48) is a functor, then weak invariance of φ is an additional structure.57 xample 9.8. If ˜ φ : ˜ C → ˜ C (cid:48) is an equivariant functor in Fun ( BG, C ∗ Cat ), then ( φ, ( id φ ))is a weakly invariant morphism from C to C (cid:48) . Here ( id φ ) denotes the constant family on id φ .If φ : C → C (cid:48) is a functor such that g − φg = φ for all g , then φ comes from a uniquelydetermined equivariant functor ˜ φ : ˜ C → ˜ C (cid:48) .Let ˜ C be in Fun ( BG, C ∗ Cat ). We say that ˜ C is α -additive if it its underlying C ∗ -category C is α -additive.Let I be a set and assume that ˜ C is | I | -additive. Then we can construct a functor (cid:76) I id C : C → C as in (9.2). Due to the choices involved in this construction we can notexpect that it comes from an invariant functor. The following lemma morally provides thebest approximation to its equivariance. Lemma 9.9.
The functor (cid:76) I id C defined by (9.2) has a canonical extension to a weaklyequivariant endofunctor ( (cid:77) I id C , θ ) : C → C . Proof.
We construct (cid:76) I id C as in Definition 9.2. It remains to construct the family ofunitary transformations θ . For g in G we consider the object gC and let ( (cid:76) I gC, ( e gi ) i ∈ I ) bethe corresponding choice of the sum. The object ( g − (cid:76) I gC, ( g − e gi ) i ∈ I ) represents a sumfor the object C . From Lemma 3.19 we get a uniquely determined unitary isomorphism θ ( g ) C : (cid:77) I C → g − (cid:77) I gC such that θ ( g ) C e i = g − e gi for all i in I . One checks that the family θ := ( θ ( g ) C ) C ∈ C is anatural transformation of functors θ ( g ) : (cid:77) I id C → g − (cid:0) (cid:77) I id C (cid:1) g which satisfies the cocycle relation required in Definition 9.6. The pair ( (cid:76) I id C , θ ) is thedesired canonical extension of ⊕ I id C to a weakly equivariant endofunctor of C .
10 Reduced crossed products
The maximal crossed product of a C ∗ -category with a strict action of a group G hasbeen introduced and studied in [Bun]. In the present paper we will introduce the reducedcrossed product. Recall that in the case of a C ∗ -algebra with G -action A , according to58efinition 10.14 the reduced norm on the algebraic crossed product ˜ A (cid:111) alg G is inducedfrom a representation on the Hilbert- C ∗ -module L ( G, ˜ A ), see (10.10) below. In the caseof C ∗ -categories we will employ G -indexed sums of objects in similar manner. The reducedcrossed product of C ∗ -categories with G -action is an important ingredient in [BE, BEL].Let ˜ C be in Fun ( BG, C ∗ Cat ). Then we have the family of functors ( g ) g ∈ G , g : C → C .We assume that ˜ C is | G | -additive. By Definition 9.2 we can consider the functor ⊕ g ∈ G g : C → C . Note that the category C ∗ Cat is complete ([Del12], [Bun19, Thm. 8.1]). In particular, itadmits pull-backs.
Definition 10.1.
We define the category L ( G, ˜ C ) as the pull-back in C ∗ CatL ( G, ˜ C ) (cid:47) (cid:47) ! (cid:15) (cid:15) C (2.3) (cid:15) (cid:15) C )] ⊕ g ∈ G g )] (cid:47) (cid:47) C )] (10.1)The category L ( G, ˜ C ) is determined uniquely up to unique unitary isomorphism. Remark 10.2.
We have the following explicit description of L ( G, ˜ C ):1. objects: The set objects of L ( G, ˜ C ) is canonically identified with the set of objectsof C using the arrow marked by ! in (10.1).2. morphisms: The definition of the sum ⊕ g ∈ G g involves the choice of an object( (cid:76) g ∈ G gC, ( e Cg ) g ∈ G ) for every object C of C . The space of morphisms from C to C (cid:48) in L ( G, ˜ C ) is then given by Hom L ( G, ˜ C ) ( C, C (cid:48) ) ∼ = Hom C (cid:0) (cid:77) g ∈ G gC, (cid:77) g ∈ G gC (cid:48) (cid:1) . (10.2)3. The composition and the involutions are inherited from C . Example 10.3.
In this example we construct for every ˜ A in Fun ( BG, C ∗ Alg nu ) an object (cid:94) Hilb G ( ˜ A ) in Fun ( BG, C ∗ Cat ).We first define a C ∗ -category Hilb G ( ˜ A ). We start with the C ∗ -category Hilb ( A ) of verysmall A -Hilbert C ∗ -modules. The action of a group element g on A will be written as a (cid:55)→ g a . 59. The objects of Hilb G ( ˜ A ) are pairs ( M, λ ) of an object M of Hilb ( A ) with a compat-ible action λ of G on M by linear maps (not necessarily A -module homomorphisms)with:a) For all a in A and m in M we have λ ( g )( m · a ) = λ ( g )( m ) · g a .b) For all m, m (cid:48) in M we have (cid:104) λ ( g )( m ) , λ ( g )( m (cid:48) ) (cid:105) M = g (cid:104) m, m (cid:105) M .2. A morphism f : ( M, λ ) → ( M (cid:48) , λ (cid:48) ) in Hilb G ( ˜ A ) is a morphism f : M → M (cid:48) in Hilb ( A ) such that f ( λ ( g )( m )) = λ (cid:48) ( g )( f ( m )) for all g in G and m in M .Note that λ ( g ) is invertible and unitary by Condition 1b.We can consider ˜ A as an object of Hilb G ( ˜ A ).We now define a strict G -action on the C ∗ -category Hilb G ( ˜ A ). An element h in G acts asan endofunctor of Hilb G ( ˜ A ) as follows:1. objects: The functor h sends an object ( M, λ ) in
Hilb G ( ˜ A ) to the object h ( M, λ ) := ( hM, hλ ) , where hM is the C -vector space M with the right multiplication by A given by( m, a ) (cid:55)→ m · h − a , the A -valued scalar product (cid:104) m, m (cid:48) (cid:105) hM := h (cid:104) m, m (cid:48) (cid:105) M , and the G -action ( hλ )( g ) := λ ( h − gh ).2. morphisms: If f : ( M, λ ) → ( M (cid:48) , λ (cid:48) ) is a morphism in Hilb G ( A ), then the morphism hf is defined as hf := f : h ( M, λ ) → h ( M (cid:48) , λ (cid:48) ).One easily checks that this G -action on Hilb G ( ˜ A ) is well-defined. We let (cid:94) Hilb G ( ˜ A ) in Fun ( BG, C ∗ Cat ) denote the category
Hilb G ( ˜ A ) with the G -action just described. Example 10.4.
This example should motivate the notation used for the C ∗ -categoryintroduced in Definition 10.1. We assume for simplicity that G acts trivially on A . Let id stand for the trivial action of G on M . Then ( M, id ) is an object of Hilb G ( ˜ A ). Inthis case by Lemma 3.23 we have an isomorphism (cid:76) g ∈ G g ( M, id ) ∼ = ( L ( G, M ) , id ) in Hilb G ( A ). We then get an isomorphism Hom L ( G, (cid:94) Hilb G ( ˜ A )) (( M, id ) , ( M (cid:48) , id )) ∼ = Hom
Hilb ( A ) ( L ( G, M ) , L ( G, M (cid:48) ))by using Remark 10.2.Let ˜ C be in Fun ( BG, C ∗ Cat ). Using the universal property of L ( G, ˜ C ) we define a60unctor σ : C → L ( G, ˜ C ) based on the following diagram: C σ (cid:36) (cid:36) (2.3) (cid:35) (cid:35) ⊕ g ∈ G g (cid:38) (cid:38) L ( G, ˜ C ) (cid:47) (cid:47) (cid:15) (cid:15) C (2.3) (cid:15) (cid:15) C )] ⊕ g ∈ G g )] (cid:47) (cid:47) C )] (10.3) Remark 10.5.
Using the explicit description of L ( G, ˜ C ) given in Remark 10.2 we cangive an explicit description of the functor σ :1. objects: In view of the left commuting triangle in (10.3) the action of σ on objectsis the identity under the identification 10.2.1.2. morphisms: Using the right commuting triangle in (10.3) and Remark 10.2.2 we seethat σ sends a morphism f : C → C (cid:48) to the morphism ⊕ g ∈ G gf : (cid:77) g ∈ G gC → (cid:77) g ∈ G gC (cid:48) in L ( G, ˜ C ). Note that one can write this also as σ ( f ) = (cid:88) g ∈ G e C (cid:48) g g ( f ) e C, ∗ g , (10.4)where ( e Cg ) g ∈ G and ( e C (cid:48) g ) g ∈ G are the isometries from the choices of sums ( (cid:76) g ∈ G gC, ( e Cg ) g ∈ G )and ( (cid:76) g ∈ G gC (cid:48) , ( e C (cid:48) g ) g ∈ G ).We recall the notion of a covariant representation of ˜ C on D in C ∗ Cat [Bun, Defn. 5.4]:
Definition 10.6.
A covariant representation of ˜ C on D is a pair ( σ, π ) consisting of:1. a functor σ : C → D
2. a family π = ( π ( g )) g ∈ G of unitary natural isomorphisms π ( g ) : σ → g ∗ σ such that g ∗ π ( g (cid:48) ) ◦ π ( g ) = π ( g (cid:48) g ) for all g, g (cid:48) in G . Lemma 10.7.
The functor σ : C → L ( G, ˜ C ) has a canonical extension to a covariantrepresentation ( σ, π ) of ˜ C on L ( G, ˜ C ) . roof. We will use the explicit descriptions of L ( G, ˜ C ) and σ given in Remarks 10.2 and10.5. We must describe π . Let h be in G . The functor h ∗ σ sends the object C of C to theobject hC in L ( G, ˜ C ), and the morphism f : C → C (cid:48) to the morphism ⊕ g ∈ G ghf : (cid:77) g ∈ G ghC → (cid:77) g ∈ G ghC (cid:48) in L ( G, ˜ C ). For every object C of C we define, applying Corollary 4.1.1 to the family( e C, ∗ gh ) g ∈ G of morphisms e C, ∗ gh : (cid:76) g ∈ G gC → ghC , the morphism π ( h ) C := (cid:88) g ∈ G e hCg e C, ∗ gh : (cid:77) g ∈ G gC → (cid:77) g ∈ G ghC . (10.5)It is straightforward to check that the family π ( h ) := ( π ( h )) C ∈ Ob( C ) is a unitary naturaltransformation from σ to h ∗ σ . Furthermore one checks that the family π := ( π ( h )) h ∈ G satisfies the cocycle condition in Definition 10.6.2.Let ˜ C be in Fun ( BG, C ∗ Cat ). According to [Bun, Defn. 5.1] we can form the algebraiccrossed product ˜ C (cid:111) alg G in ∗ Cat C . Instead of repeating the definition of the crossed product we proceed, assumingthat ˜ C is | G | -additive, with observing that by [Bun, Lem. 5.7] the covariant representation( σ, π ) from Lemma 10.7 induces a functor ρ : ˜ C (cid:111) alg G → L ( G, ˜ C ) . (10.6)In our situation this functor is wide and faithful, and we can describe the algebraic crossedproduct ˜ C (cid:111) alg G directly as a C -linear ∗ -subcategory of L ( G, ˜ C ):1. objects: The objects of ˜ C (cid:111) alg G are the objects of C and hence of L ( G, ˜ C ).2. morphisms: The C -vector space of morphisms Hom ˜ C (cid:111) alg G ( C, C (cid:48) ) is linearly generatedas a subspace of
Hom L ( G, ˜ C ) ( C, C (cid:48) ) by the morphisms π ( g ) g − C (cid:48) σ ( f ) for all g in G and f : C → g − C (cid:48) in C .3. The composition and the involution are inherited from L ( G, ˜ C ).One easily checks that this describes a well-defined subcategory which is equivalent to thealgebraic crossed product ˜ C (cid:111) alg G defined in [Bun, Defn. 5.1].Using the notation ( f, g ) for a morphism C → C (cid:48) in C (cid:111) alg G (where f : C → g − C (cid:48) ) asin [Bun, Defn. 5.1] we have (using [Bun, (5.4)] for the first equality) ρ ( f, g ) = π ( g ) g − C (cid:48) σ ( f ) (10.5) , (10.4) = (cid:88) (cid:96) ∈ G e C (cid:48) (cid:96) e g − C (cid:48) , ∗ (cid:96)g (cid:88) (cid:96) (cid:48) ∈ G e g − C (cid:48) (cid:96) (cid:48) (cid:96) (cid:48) ( f ) e C, ∗ (cid:96) (cid:48) = (cid:88) (cid:96) ∈ G e C (cid:48) (cid:96) ( (cid:96)g ) f e C, ∗ (cid:96)g , (10.7)62here we use the orthogonality relations for the family ( e g − C (cid:48) (cid:96) ) (cid:96) ∈ G for the last equality.Let ˜ C be a | G | -additive object of Fun ( BG, C ∗ Cat ). Definition 10.8.
The reduced crossed product ˜ C (cid:111) r G is the closure of ˜ C (cid:111) alg G withrespect to the norm induced by the representation ρ . Equivalently, ˜ C (cid:111) r G is the closure of ˜ C (cid:111) alg G viewed as a subcategory of L ( G, ˜ C ). Example 10.9.
We continue with Example 10.3. We consider ˜ A as an object of (cid:94) Hilb G ( ˜ A ).If A is unital, then we have an isomorphism of C ∗ -algebras End (cid:94)
Hilb G ( ˜ A ) (cid:111) r G ( ˜ A ) ∼ = ˜ A (cid:111) r G , where the right-hand side is the classical reduced crossed product of A with G . Unitalityof A is needed to ensure that the set of all bounded adjointable operators on the A -Hilbert C ∗ -module A is exactly A acting by left multiplication. In this sense Definition 10.8generalizes the reduced crossed product from C ∗ -algebras with G -action to C ∗ -categorieswith G -action.Recall the notation introduced in Section 7. In particular, (cid:92) C ∗ Cat | G | add is the subcategoryof C ∗ Cat of | G | -additive categories and | G | -additive functors. Lemma 10.10.
The construction of the reduced crossed product has a canonical extensionto a functor − (cid:111) r G : Fun ( BG, (cid:92) C ∗ Cat | G | add ) → C ∗ Cat . Proof.
We have already described the action of the functor on objects. We must extend itto morphisms. Thus let ˜ φ : ˜ C → ˜ C (cid:48) be a morphism in Fun ( BG, (cid:92) C ∗ Cat | G | add ). It inducesa functor ˜ φ (cid:111) alg G : ˜ C (cid:111) alg G → ˜ C (cid:48) (cid:111) alg G in ∗ Cat C in a functorial way. We must show that it extends by continuity to the reducedcrossed products.The pull-back square in (10.1) is also a pull-back square in the (2 , C ∗ Cat , ofunital C ∗ -categories, unital functors and unitary natural isomorphisms. The functor ˜ φ preserves the functor ⊕ g ∈ G g up to unique unitary isomorphism by assumption. It followsthat the functor ˜ φ induces a morphism of pull-back squares of the shape (10.1) in C ∗ Cat , ,and therefore a functor L ( G, ˜ φ ) : L ( G, ˜ C ) → L ( G, ˜ C (cid:48) )63hich is uniquely determined up to unitary isomorphism. Furthermore, in view of theconstruction of σ in (10.3) we get a diagram C φ (cid:47) (cid:47) σ (cid:15) (cid:15) C (cid:48) σ (cid:48) (cid:15) (cid:15) L ( G, ˜ C ) L ( G, ˜ φ ) (cid:47) (cid:47) L ( G, ˜ C (cid:48) )which commutes up to unitary isomorphism. One checks using the explicit descriptions that L ( G, ˜ φ ) restricts to the algebraic crossed products viewed as subcategories of L ( G, ˜ C )and L ( G, ˜ C (cid:48) ), respectively, and that this restriction is equivalent to ˜ φ (cid:111) alg G . Thus wecan now define ˜ φ (cid:111) r G : ˜ C (cid:111) r G → ˜ C (cid:48) (cid:111) r G as the continuous extension of ˜ φ (cid:111) alg G .One checks in a straightforward manner that − (cid:111) r G is compatible with the composition. Remark 10.11.
Assume that ˜ C is in Fun ( BG, C ∗ Cat ) and | G | -additive. Assume furtherthat ˜ D is a G -invariant subcategory of ˜ C . Abusing notation, we will sometimes (seeRemark 17.16) define ˜ D (cid:111) r G as the subcategory of ˜ C (cid:111) r G obtained as closure of thesubcategory ˜ D (cid:111) alg G inside ˜ C (cid:111) r G . If ˜ D is a full subcategory of ˜ C , then ˜ D (cid:111) r G is the fullsubcategory of ˜ C (cid:111) r G on the objects of D . A priori, ˜ D (cid:111) r G depends on the surrounding | G | -additive category ˜ C . In order to indicate this dependence we will occasionally use thenotation ˜ D (cid:111) ˜ C r G .In Lemma 10.12 we will see that under natural conditions ˜ D (cid:111) ˜ C r G is independent of ˜ C .Let ˜ C and ˜ C be | G | -additive objects in Fun ( BG, C ∗ Cat ), and let ˜ D be included as afull G -invariant subcategory into ˜ C and ˜ C . Lemma 10.12.
There is a canonical isometric ∗ -isomorphism Φ : ˜ D (cid:111) ˜ C r G → ˜ D (cid:111) ˜ C r G .Proof. By definition the sets of objects of ˜ D (cid:111) ˜ C r G and ˜ D (cid:111) ˜ C r G are equal to the set ofobjects of D . To compare the morphisms, let D, D (cid:48) be objects of D . We first show thatthere exists a canonical isometric isomorphismΦ D,D (cid:48) : Hom C (cid:0) (cid:77) g ∈ G gD, (cid:77) g ∈ G gD (cid:48) (cid:1) → Hom C (cid:0) (cid:77) g ∈ G gD, (cid:77) g ∈ G gD (cid:48) (cid:1) , (10.8)cf. (10.2). To this end let φ : (cid:76) g ∈ G gD → (cid:76) g ∈ G gD (cid:48) be a morphism in C . By Corollary 4.1it is determined by a unique family of morphisms ( φ kl ) k,l ∈ G in C with φ kl : kD → lD (cid:48) .Since ˜ D is a full and G -invariant subcategory of both ˜ C and of ˜ C , the morphisms φ kl for each k and l are also morphisms of D and hence of C . By the Corollary 4.1, nowapplied in C , the family ( φ kl ) k,l ∈ G defines a unique morphism φ (cid:48) : (cid:76) g ∈ G gD → (cid:76) g ∈ G gD (cid:48) C . The assignment φ (cid:55)→ φ (cid:48) defines Φ D,D (cid:48) , and by Proposition 3.22 together with thecomputation of the norm of multipliers from Lemma 3.14 we conclude that Φ is isometric.It is clear that Φ
D,D (cid:48) is bijective (one constructs the inverse morphism analogously).By similar arguments one concludes that the family of isomorphisms (Φ
D,D (cid:48) ) D,D (cid:48) ∈ D iscompatible with composition of morphisms and with the involution.The family (Φ D,D (cid:48) ) D,D (cid:48) ∈ D induces the canonical isometric ∗ -isomorphism Φ as claimed bythe lemma.Recall the functor A from (4.1) which is explained in details in Remark 4.5. Let ˜ C be in Fun ( BG, C ∗ Cat ) and consider A ( ˜ C ) in Fun ( BG, C ∗ Alg nu ). We have an isomorphism A alg ( ˜ C (cid:111) alg G ) ∼ = A alg ( ˜ C ) (cid:111) alg G . (10.9)In [Bun, Thm. 6.9] we have shown that this isomorphism extends to an isomorphism A ( ˜ C (cid:111) G ) ∼ = A ( ˜ C ) (cid:111) G involving maximal crossed products. The main result here is the analogue for the reducedcrossed products.We assume that C is | G | -additive. Theorem 10.13.
The isomorphism (10.9) extends to an isomorphism A ( ˜ C (cid:111) r G ) ∼ = A ( ˜ C ) (cid:111) r G .
We start with the explicit description of the reduced crossed product of C ∗ -algebras with G -action. Let ˜ A be in Fun ( BG, C ∗ Alg nu ). We write the action of g on a in A as g a .Let M be in Hilb ( A ). We let · and (cid:104)− , −(cid:105) denote the right A -module structure and the A -valued scalar product of M . For g in G we form the object g M of Hilb ( A ) given asfollows:1. underlying C -vector space: g M := M
2. right module structure: m · g a := m · g − a
3. scalar product: (cid:104)− , − (cid:48) (cid:105) g := g (cid:104)− , − (cid:48) (cid:105) Note that ( g, M ) (cid:55)→ g M and ( g, f ) (cid:55)→ g f := f is an action of G on Hilb ( A ). We let Hilb ( ˜ A ) be the corresponding object of Fun ( BG, C ∗ Cat ).65e consider A as an object of Hilb ( ˜ A ) in the natural way and define L ( G, ˜ A ) := (cid:77) g ∈ G g − A (10.10)in Hilb ( A ). An element of the summand with index g will be denoted by [ g, b ], where b isin g A = A . We define a covariant representation ( ρ, π ) of ( A, G ) on L ( G, ˜ A ) as follows:1. For a in A we define ρ ( a ) in End
Hilb ( A ) ( L ( G, ˜ A )) such that ρ ( a )[ g, b ] = [ g, ab ].2. For h in G we define a unitary π ( h ) in End
Hilb ( A ) ( L ( G, ˜ A )) by π ( h )([ g, b ]) := [ hg, h b ].One checks the relation π ( h ) ρ ( a ) π ( h − ) = ρ ( h a ).Recall that we have the following definition. Let ˜ A be in Fun ( BG, C ∗ Alg nu ). Definition 10.14.
The reduced crossed product ˜ A (cid:111) r G of ˜ A by G is the C ∗ -subalgebra of End
Hilb ( A ) ( L ( G, ˜ A )) generated by the operators π ( g ) ρ ( a ) for all in A and h in G . If ˜ C is in Fun ( BG, C ∗ Cat ), then A ( ˜ C ) is in Fun ( BG, C ∗ Alg nu ). Hence we get Hilb ( A ( ˜ C ))in Fun ( BG, C ∗ Cat ). Lemma 10.15.
The functor M : C → Hilb ( A ( ˜ C )) from (4.6) is a morphism in Fun ( BG, C ∗ Cat ) .Proof. We have an isomorphism g M C = g (1 C A ( C )) m (cid:55)→ g m ∼ = 1 gC A ( C ) = M gC as vector spaces. One checks that this isomorphism intertwines the right A ( C )-action · on M gC with the action · g on g M C . Finally it is an isometry if we equip g M C with the scalarproduct g (cid:104)− , − (cid:48) (cid:105) M C .If f : C → C (cid:48) is a morphism in C , then g M C g f (cid:47) (cid:47) (cid:15) (cid:15) g M C (cid:48) (cid:15) (cid:15) M gC M gf (cid:47) (cid:47) M gC (cid:48) commutes. 66e consider C be in C ∗ Cat and A ( C ) in C ∗ Alg nu . As before we consider A ( C ) as an objectin Hilb ( A ( C )). Note that the algebraic sum (cid:76) alg C ∈ C M C is naturally a A ( C )-submodule of A ( C ). Lemma 10.16.
We have an isomorphism A ( C ) ∼ = (cid:76) C ∈ C M C in Hilb ( A ( C )) .Proof. We use that (cid:76) alg C ∈ C M C is a dense subspace of both A ( C ) and (cid:76) C ∈ C M C . Let m = ⊕ C m C be an element of (cid:76) alg C ∈ C M C . Then (cid:107) m (cid:107) A ( C ) = (cid:107) m ∗ m (cid:107) A ( C ) = (cid:104) m ∗ , m (cid:105) (cid:76) C ∈ C M C = (cid:107) m (cid:107) (cid:76) C ∈ C M C . This equality of norms implies the equality of closures.We have an isomorphism in
Hilb ( A ( C )) L ( G, A ( ˜ C )) (10.10) ∼ = (cid:77) g ∈ G (cid:77) C ∈ C g − M C Lem. 10 . ∼ = (cid:77) g ∈ G (cid:77) C ∈ C M gC . (10.11)Under this isomorphism the operator ρ ( f [ C (cid:48) , C ]) goes to the operator given by˜ ρ ( f [ C (cid:48) , C ])[ g, b ] := [ g, ( gf )[ gC (cid:48) , gC (cid:48)(cid:48) ] b ] (10.12)and the unitary π ( h ) goes to the unitary given by˜ π ( h )[ g, b ] := [ hg, h b ] . (10.13)Using the functor M from (4.6) and Lemma 10.15 we embed ˜ C into Hilb ( A ( ˜ C )). In viewof (10.2) and Corollary 4.7 the C ∗ -category L ( G, ˜ C ) from Definition 10.1 is equivalent tothe full subcategory of Hilb ( A ( C )) on the objects (cid:76) g ∈ G M gC for all C in C . So Hom L ( G, ˜ C ) ( C, C (cid:48) ) ∼ = Hom
Hilb ( A ( C )) (cid:0) (cid:77) g ∈ G M gC , (cid:77) g ∈ G M gC (cid:48) (cid:1) . (10.14)Under this isomorphism the covariant representation of ˜ C on L ( G, ˜ C ) from Lemma 10.7sends f : C → C (cid:48) in C to the operator σ ( f ) in Hom
Hilb ( A ( C )) ( (cid:76) g ∈ G M gC , (cid:76) g ∈ G M gC (cid:48) ) givenby σ ( f )[ g, b ] := [ g, ( gf )[ gC (cid:48) , gC ] b ] (10.15)(where b in M gC is considered as an element of A ( C ) so that ( gf )[ gC (cid:48) , gC ] b belongs to M gC again considered as a subspace of A ( C )), and h in G to the unitary operator in Hom
Hilb ( A ( C )) ( (cid:76) g ∈ G M gC , (cid:76) g ∈ G M gC ) given by κ ( h )[ g, b ] := [ hg, h b ] . (10.16)By Definition 10.8 the reduced crossed product C (cid:111) r G is the wide subcategory of L ( G, ˜ C )generated by the image of this covariant representation.67 roof of Theorem 10.13. We have an isomorphism A alg ( L ( G, ˜ C )) ∼ = (cid:77) C,C (cid:48)
Hom
Hilb ( A ( C )) (cid:0)(cid:77) g ∈ G M gC , (cid:77) g (cid:48) ∈ G M g (cid:48) C (cid:48) (cid:1) . (10.17)We furthermore have a canonical homomorphism A alg ( L ( G, ˜ C )) → End
Hilb ( A ( C )) ( L ( G, A ( C ))) . (10.18)We claim that the homomorphism (10.18) sends the linear generators of A ( C (cid:111) r G ) to thelinear generators of A ( C ) (cid:111) r G .To this end we first describe explicitly the action of (10.18) on linear generators under theisomorphisms (10.11) and (10.17). We consider an element of A alg ( L ( G, ˜ C )) describedby a one-entry matrix F [ C (cid:48) , C ] in Hom
Hilb ( A ( C )) ( (cid:76) g ∈ G M gC , (cid:76) g (cid:48) ∈ G M g (cid:48) C (cid:48) ). This operatoris given by a matrix ( F [ C (cid:48) , C ] g (cid:48) ,g ) g,g (cid:48) ∈ G , where F [ C (cid:48) , C ] g (cid:48) ,g = M f [ C,C (cid:48) ] g (cid:48) ,g with f [ C (cid:48) , C ] g (cid:48) ,g : gC → g (cid:48) C (cid:48) . The image of F [ C (cid:48) , C ] in End
Hilb ( A ( C )) ( L ( G, A ( C ))) corresponds to the operator in End
Hilb ( A ( C )) ( (cid:76) g ∈ G (cid:76) C ∈ C M gC ) which sends [ g, b ] with b in M C (cid:48)(cid:48) to zero if C (cid:54) = C (cid:48)(cid:48) and to (cid:80) g (cid:48) ∈ G [ g (cid:48) , F [ C (cid:48) , C ] g (cid:48) ,g b ] if C = C (cid:48)(cid:48) .In particular for f : C → C (cid:48) in C the operator σ ( f )[ C (cid:48) , C ] in A alg ( L ( G, ˜ C )) (see (10.15)) issent to the operator given by the diagonal matrix diag (( gf ) g ∈ G ). The operator ˜ ρ ( f [ C (cid:48) , C ])from (10.12) is given by the same diagonal matrix. Similarly, for h in G the operator κ ( h ) ˜ ρ (1 C [ C, C ]) (see (10.16)) is given by the matrix F [ h − C, C ], where F [ h − C, C ] g (cid:48) ,g iszero if g (cid:48) h − (cid:54) = g and 1 gC if g (cid:48) h − = g . The operator ˜ π ( h ) ˜ ρ (1 C ) from (10.13) and (10.12)is given by the same matrix. This finishes the proof of the claim.The claim implies now the Theorem 10.13. Remark 10.17.
In order to define the reduced crossed product we assumed in Definition10.8 that C is | G | -additive. We can use the embedding ˜ M : ˜ C → Hilb ( A ( ˜ C )) in order toremove this assumption. Using the notation from Remark 10.11 we could define for any ˜ C in Fun ( BG, C ∗ Cat ) ˜ C (cid:111) r G := ˜ C (cid:111) Hilb ( A ( ˜ C )) r G .
It follows from Lemma 10.12 that this definition is equivalent to Definition 10.8 in thecase that ˜ C itself is | G | -additive.In the following we compare the reduced and the maximal versions of the crossed product.Let ˜ C be in Fun ( BG, C ∗ Cat ). By Definition 10.8 the norm on the reduced crossed product˜ C (cid:111) r G is induced by the representation ρ from (10.6) which comes from the covariant68epresentation ( σ, π ) from Lemma 10.7. Hence by the universal property of the maximalcrossed product [Bun, Cor. 5.10] we get a comparison functor q ˜ C : ˜ C (cid:111) G → ˜ C (cid:111) r G (10.19)in C ∗ Cat . Lemma 10.18.
The functor q ˜ C is the identity on objects and surjective on morphismspaces.Proof. By construction, q ˜ C is the identity on objects.By definition of the reduced crossed product, the image of the functor q ˜ C contains thedense ∗ -subcategory ˜ C (cid:111) alg G of ˜ C (cid:111) r G . Since functors between C ∗ -categories have closedranges on the morphism spaces the claim follows.It is known that for an amenable group G the canonical map q ˜ A : ˜ A (cid:111) G → ˜ A (cid:111) r G is an isomorphism for all ˜ A in Fun ( BG, C ∗ Alg nu ). In the following we generalize this factto C ∗ -categories.Let ˜ C be in Fun ( BG, C ∗ Cat ). Theorem 10.19. If G is amenable, then the canonical functor q ˜ C : ˜ C (cid:111) G → ˜ C (cid:111) r G isan isomorphism.Proof. Since for any C ∗ -category D the canonical map ρ D : D → A ( D ) is an isometry[Bun, Lem. 6.7] it suffices to show that A ( q ˜ C ) : A ( ˜ C (cid:111) G ) → A ( ˜ C (cid:111) r G ) is an isomorphism.Recall that the isomorphism from (10.9) extends to isomorphisms A ( ˜ C ) (cid:111) G ∼ = −→ A ( ˜ C (cid:111) G ) (10.20)by [Bun, Thm. 6.9], and A ( ˜ C ) (cid:111) r G ∼ = −→ A ( ˜ C (cid:111) r G ) (10.21)by Theorem 10.13. We have a commutative diagram A ( ˜ C ) (cid:111) G (10.20) ∼ = (cid:47) (cid:47) q A ( ˜ C ) (cid:15) (cid:15) A ( ˜ C (cid:111) G ) A ( q ˜ C ) (cid:15) (cid:15) A ( ˜ C ) (cid:111) r G (10.21) ∼ = (cid:47) (cid:47) A ( ˜ C (cid:111) r G ) (10.22)The left vertical arrow q A ( ˜ C ) is an isomorphism, because G is amenable and A ( ˜ C ) is a C ∗ -algebra. This implies that A ( q ˜ C ) is an isomorphism, too.69 We first introduce the notions of a closed ideal in a C ∗ -category and of an excisive squareof C ∗ -categories. Then we define the notion of a homological functor and discuss some ofits basic properties.Let i : C → D be a functor in C ∗ Cat nu . The following definition is equivalent to [Bun,Def. 8.2]. It generalizes the notion of a closed two-sided ideal in a C ∗ -algebra. Definition 11.1.
The functor i is an inclusion of an ideal (is a kernel) if it has thefollowing properties:1. i induces a bijection between the sets of objects.2. i induces closed embeddings of morphism spaces.3. The composition of a morphism in the image of i with any morphism of D belongsagain to the image of i . Let i : C → D be a functor in C ∗ Cat nu . Since the category C ∗ Cat nu is cocomplete [Bun,Thm. 4.1.2.] we can define the quotient D / C in C ∗ Cat nu by the push-out C i (cid:47) (cid:47) (2.3) (cid:15) (cid:15) D (cid:15) (cid:15) C )] (cid:47) (cid:47) D / C If i is the inclusion of an ideal it is easy to describe the C ∗ -category D / C explicitly.1. objects: The objects of D / C are the objects of D (which are in bijection with theobjects of C via i ).2. morphisms: For objects C, C (cid:48) in C we have Hom D / C ( i ( C ) , i ( C (cid:48) )) ∼ = Hom D ( i ( C ) , i ( C (cid:48) )) /i ( Hom C ( C, C (cid:48) )) .
3. composition and involution: The composition and ∗ -operation are inherited from D .Since i ( Hom C ( C, C (cid:48) )) is a closed subspace of
Hom D / C ( i ( C ) , i ( C (cid:48) )) the quotient has aninduced norm which exhibits D / C as a C ∗ -category [Mit02, Cor. 4.8]. If D is unital, thenso is D / C , and the projection map D → D / C is a morphism in C ∗ Cat .70e consider a square A (cid:47) (cid:47) (cid:15) (cid:15) B (cid:15) (cid:15) C (cid:47) (cid:47) D (11.1)in C ∗ Cat nu . By the universal property of the quotients of the horizontal functors weobtain an induced functor B / A → D / C .The following is taken from [Bun, Defn. 8.10]. Definition 11.2.
The square (11.1) is called excisive if it satisfies the following conditions:1. The functors A → B and C → D are embeddings of closed ideals.2. The quotients B / A and D / C are unital3. The induced functor B / A → D / C is unital and a unitary equivalence. Example 11.3.
The notion of an excisive square of C ∗ -categories generalizes the notionof an exact sequence of C ∗ -algebras in the following sense. Let0 → I → A → Q → C ∗ -algebras where I is a closed ideal in A and Q is unital.We can consider C ∗ -algebras as C ∗ -categories with a single object ∗ . Then the square I (cid:47) (cid:47) (cid:15) (cid:15) A (cid:15) (cid:15) ∗ ] (cid:47) (cid:47) Q is an excisive square of C ∗ -categories. Remark 11.4.
In the following we use the language of ∞ -categories . References are[Lur09, Cis19]. Ordinary categories will be considered as ∞ -categories using the nervefunctor. A typical target ∞ -category for the homological functors introduced below isthe stable ∞ -category Sp of spectra. We refer to [Lur] for an introduction to stable ∞ -categories in general, and for Sp in particular. The ∞ -categories considered in thepresent paper belong to the large universe. A cocomplete ∞ -category thus admits allcolimits for small index categories.Let M be an ∞ -category. We consider a functorHg : C ∗ Cat nu → M . more precisely, ( ∞ , efinition 11.5. Hg is a homological functor if the following conditions are satisfied:1. M is stable.2. Hg sends unitary equivalences between unital C ∗ -categories to equivalences.3. Hg sends excisive squares to push-out squares.4. For every small set X we have Hg(0[ X ]) (cid:39) M . Let Hg : C ∗ Cat nu → M be a homological functor. Definition 11.6. Hg is finitary if M is in addition cocomplete and Hg preserves smallfiltered colimits. Let f : C → D be a functor in C ∗ Cat nu . Definition 11.7. f is called a zero functor if it sends every morphism to zero. Let Hg : C ∗ Cat nu → M be a homological functor, and let f : C → D be a functor in C ∗ Cat nu . Lemma 11.8. If f is a zero functor, then Hg( f ) = 0 .Proof. The functor f has an obvious factorization C → D )] ω D −−→ D , where ω D is the counit of the adjunction in [Bun, (3.12)]: it acts as the identity on objectsand in the only possible way on morphisms. By functoriality of Hg we get a factorizationof Hg( f ) as Hg( C ) → Hg(0[Ob( D )]) → Hg( D ) . The assertion follows from Condition 11.5.4 on Hg which implies Hg(0[Ob( D )]) (cid:39) M .Let Hg : C ∗ Cat nu → M be a homological functor. Lemma 11.9.
Assume:1. C and D are unital. . C and D admit zero objects.3. Hg is homological.Then the morphism (Hg( pr C ) , Hg( pr D )) : Hg( C × D ) → Hg( C ) × Hg( D ) is an equivalence.Proof. We consider the following square0[Ob( C )] ω C (cid:47) (cid:47) (cid:15) (cid:15) C z C (cid:15) (cid:15) C )] × D ω C × id D (cid:47) (cid:47) C × D (11.2)in C ∗ Cat nu . The functor ω C was explained in the proof of Lemma 11.8. The functor z C sends an object C of C to the object ( C, D ) of C × D , where 0 D is a chosen zero object in D , and it sends a morphism f : C → C (cid:48) in C to the morphism ( f,
0) : ( C, D ) → ( C (cid:48) , D )in C × D . The left vertical functor is defined analogously.We claim that the square (11.2) is excisive. The horizontal functors in (11.2) are inclusionsof closed ideals. The quotient of the upper horizontal functor is canonically isomorphicto C . The quotient of the lower horizontal functor is isomorphic to C × D )] via theobvious projection. The induced functor i C : C → C × D )] between the quotients isgiven as follows:1. objects: i C sends an object C of C to the object ( C, D ) in C × D )].2. morphisms: i C sends a morphism f : C → C (cid:48) in C to the morphism ( f,
0) : ( C, D ) → ( C (cid:48) , D ) in C × D )].Both quotients are unital, and the functor i C is unital.The functor i C is a unitary equivalence. An inverse functor is given by the projection p C : C × D )] → C . It is clear that p C ◦ i C = id C . It remains to describe a unitaryisomorphism u : id C × D )] → i C ◦ p C . For an object ( C, D ) in C × D )] we definethe unitary u ( C,D ) := ( id C ,
0) : (
C, D ) → ( C, D ) in C × D )]. Then the desiredunitary isomorphism is u = ( u ( C,D ) ) ( C,D ) ∈ Ob( C × D )]) . This finishes the verification thatthe square (11.2) is excisive.Since Hg is homological, by Condition 11.5.3 we get a cocartesian squareHg(0[Ob( C )]) (cid:47) (cid:47) (cid:15) (cid:15) Hg( C ) Hg( z C ) (cid:15) (cid:15) Hg(0[Ob( C )] × D ) Hg( ω C × id D ) (cid:47) (cid:47) Hg( C × D ) (11.3)in M . 73imilarly as above we have a unitary equivalence i D : D → C )] × D between unital C ∗ -algebras which sends an object D in D to the pair (0 C , D ) in 0[Ob( C )] × D , where 0 C is a chosen zero object in C . We let z D := ( ω C × id D ) ◦ i D : D → C × D . Using Conditions 11.5.4 and 11.5.2 of Hg we see that (11.3) is equivalent to the cocartesiansquare 0 M (cid:47) (cid:47) (cid:15) (cid:15) Hg( C ) Hg( z C ) (cid:15) (cid:15) Hg( D ) Hg( z D ) (cid:47) (cid:47) Hg( C × D ) (11.4)We conclude that Hg( z C ) + Hg( z D ) : Hg( C ) ⊕ Hg( D ) → Hg( C × D ) (11.5)is an equivalence. Here we also use the stability condition from Definition 11.5.1 on M in order to identify coproducts with products which are then written as sums. We nowobserve that pr C ◦ z C = id C and pr D ◦ z D = id D , and that pr C ◦ z D and pr C ◦ z D arezero morphisms. This immediately implies thatHg( pr C ) ⊕ Hg( pr D ) : Hg( C × D ) → Hg( C ) ⊕ Hg( D )is an equivalence inverse to (11.5).Our next result asserts that a homological functor is additive on functors. We assume that C , D are in C ∗ Cat , that C admits a zero object, and that D is additive. If φ, φ (cid:48) : C → D are two functors, then we can define a functor φ ⊕ φ (cid:48) : C → D by Definition 9.2.Since M is stable its morphism spaces are group-like abelian monoids in Spc . The symbol+ in the following proposition is induced by this structure.
Proposition 11.10. If Hg is a homological functor, then we have an equivalence Hg( φ ⊕ φ (cid:48) ) (cid:39) Hg( φ ) + Hg( φ (cid:48) ) : Hg( C ) → Hg( D ) . Proof.
Since D is additive we can choose a zero object 0 in D . We consider the diagramHg( D × D ) Hg( (cid:76) ) (cid:38) (cid:38) (cid:39) Hg( pr ) ⊕ Hg( pr ) (cid:118) (cid:118) Hg( D ) ⊕ Hg( D ) + (cid:47) (cid:47) Hg( D ) (11.6)74here the left vertical morphism is an equivalence by Lemma 11.9. We claim that (11.6)naturally commutes. Using the explicit inverse (11.5) of Hg( pr ) ⊕ Hg( pr ) and theuniversal property of + it suffices to show that the compositionsHg( D ) ι i → Hg( D ) ⊕ Hg( D ) Hg( z )+Hg( z ) −−−−−−−−→ Hg( D × D ) Hg( (cid:76) ) −−−−→ Hg( D )are equivalent to the identity, where ι i : Hg( D ) ι i −→ Hg( D ) ⊕ Hg( D ) denote the canonicalinclusions for i = 0 , i = 0 this composition is induced by applying Hg to the endofunctor s : D → D which is given as follows:1. objects: s sends an object D to the representative D ⊕ (cid:76) .2. morphisms: s sends a morphism f : D → D (cid:48) to the morphism f ⊕ D ⊕ → D (cid:48) ⊕ u : id D → s given by the family ( u D ) D ∈ Ob( D ) of the canonicalinclusions u D : D → D ⊕
0. Hence Hg( s ) (cid:39) Hg( id D ). The case i = 1 is analoguous.We have the following diagram in M Hg( C ) diag Hg( C ) (cid:39) (cid:39) Hg( diag C ) (cid:120) (cid:120) Hg( C × C ) (cid:39) Hg( pr ) ⊕ Hg( pr ) (cid:47) (cid:47) Hg( φ × φ (cid:48) ) (cid:15) (cid:15) Hg( C ) ⊕ Hg( C ) Hg( φ ) ⊕ Hg( φ (cid:48) ) (cid:15) (cid:15) Hg( D × D ) Hg( (cid:76) ) (cid:38) (cid:38) (cid:39) Hg( pr ) ⊕ Hg( pr ) (cid:47) (cid:47) Hg( D ) ⊕ Hg( D ) + (cid:119) (cid:119) Hg( D ) (11.7)The lower triangle is (11.6) and commutes as shown above. The upper triangle and themiddle square obviously commute. The left top-down path is the map Hg( φ ⊕ φ (cid:48) ), whilethe right top-down path is Hg( φ ) + Hg( φ (cid:48) ). The filler of (11.7) now provides the desiredequivalence between these morphisms.Since the operation + occuring in Proposition 11.10 is abelian we immediately get thefollowing consequence. Corollary 11.11.
Hg( φ ⊕ φ (cid:48) ) is equivalent to Hg( φ (cid:48) ⊕ φ ) . Recall the notion of a flasque C ∗ -category introduced in Definition 9.3.75 roposition 11.12. A homological functor annihilates flasques.Proof.
Let Hg be a homological functor. Furthermore, let C be in C ∗ Cat and assumethat it is flasque. We must show that Hg( C ) (cid:39) S : C → C implements the flasqueness of C . Then using Proposition 11.10we have the relation Hg( S ) = Hg( id ⊕ S ) = id Hg( C ) + Hg( S )in the abelian group [Hg( C ) , Hg( C )]. This implies that Hg( C ) (cid:39)
12 Topological K -theory of C ∗ -categories The goal of this section is to provide a reference for the topological K -theory functorfor C ∗ -categories. Most of the material is from [Joa03]. The main result (Theorem 12.4)states that this K -theory functor is a finitary homological functor (Definitions 11.5 and11.6).Our starting point is the topological K -theory functor K C ∗ for C ∗ -algebras. Let C ∗ Alg nu denote the category of very small possibly non-unital C ∗ -algebras and not necessarily unit-preserving homomorphisms. One could view C ∗ Alg nu as a full subcategory of C ∗ Cat nu consisting of the C ∗ -categories with a single object. Topological K -theory of C ∗ -algebrasis a functor K C ∗ : C ∗ Alg nu → Sp . References for the induced group-valued functor π ∗ K C ∗ : C ∗ Alg nu → Ab Z / Z gr (whose construction predates the spectrum-valued version) are, e.g. [Bla98, HR00], whilethe spectrum-valued one is defined in [Joa03, Defn. 4.9] and justified by [Joa03, Thm. 4.10].An alternative construction using spectrum-valued KK -theory can be based on [LN18],see also [BEL].In the following we list all the properties which will be explicitly used in the proof ofTheorem 12.4 below. Proposition 12.1.
The functor K C ∗ has the following properties.1. K C ∗ (0) (cid:39) .2. K C ∗ preserves filtered colimits. . K C ∗ sends exact sequences of C ∗ -algebras → A → B → C → to cocartesiansquares K C ∗ ( A ) (cid:47) (cid:47) (cid:15) (cid:15) K C ∗ ( B ) (cid:15) (cid:15) (cid:47) (cid:47) K C ∗ ( C ) in Sp .4. K C ∗ is K -stable (see Remark 12.2.2).5. K C ∗ is homotopy invariant (see Remark 12.2.3).6. K C ∗ is Bott periodic (see Remark 12.2.4). Remark 12.2.
In this remark we add some details to the statement of Proposition 12.1.1. An exact sequence of C ∗ -algebras as in 12.1.3 is a square A (cid:47) (cid:47) (cid:15) (cid:15) B (cid:15) (cid:15) (cid:47) (cid:47) C in C ∗ Alg nu which is a pull-back and a push-out at the same time. Assertion 12.1.3says that K C ∗ sends this square to a cocartesian (or equivalently by stability of Sp ,to a cartesian) square in Sp .2. K -stability: Let K denote the C ∗ -algebra of compact operators on a separable Hilbertspace. Fixing a rank-one projection p in K we get a morphism C → K , λ (cid:55)→ λp , in C ∗ Alg nu . For every C ∗ -algebra A we get an induced morphism A ∼ = A ⊗ C → A ⊗ K (all choices of a C ∗ -algebraic tensor product coincide in this case). Stability thensays that the induced map of spectraK C ∗ ( A ) → K C ∗ ( A ⊗ K )is an equivalence.3. homotopy invariance: For every C ∗ -algebra A the map A → C ([0 , , A ) given bythe inclusion of A as constant functions induces an equivalenceK C ∗ ( A ) → K C ∗ ( C ([0 , , A )) .
4. Bott periodicity: For every C ∗ -algebra A we have a natural equivalenceΣ K C ∗ ( A ) (cid:39) K C ∗ ( A ) .
77s observed by J. Cuntz this property is actually a formal consequence of the otherproperties stated in Proposition 12.1.In view of Bott periodicity, in order to show that a morphism K C ∗ ( A ) → K C ∗ ( B ) isan equivalence it suffices to show that π i K C ∗ ( A ) → π i K C ∗ ( B ) is an isomorphism for i = 0 , C ∗ -algebras into C ∗ -categories is the right-adjoint of an adjunction A f : C ∗ Cat nu (cid:28) C ∗ Alg nu : incl . (12.1)We refer to [Bun, Lem. 3.9] for details. Note that the functor A f has been first introducedin [Joa03]. Following [Joa03] we adopt the following definition. Definition 12.3.
We define the topological K -theory functor for C ∗ -categories as thecomposition K C ∗ Cat : C ∗ Cat nu A f −→ C ∗ Alg nu K C ∗ −−→ Sp . Note that Mitchener [Mit01] provided an alternative construction of a K -theory functorfor C ∗ -categories.For the following theorem recall Definitions 11.5 and 11.6. Theorem 12.4.
The functor K C ∗ Cat is a finitary homological functor.Proof.
This follows from Lemma 12.5, Proposition 12.6, Proposition 12.9, and Lemma12.10. All these results will be shown below.
Lemma 12.5.
The functor K C ∗ Cat preserves filtered colimits.Proof.
By definition, the functor A f is a left-adjoint and therefore preserves all colimits.The functor K C ∗ preserves filtered colimits by Proposition 12.1.2. Hence the compositionK C ∗ Cat preserves filtered colimits.
Proposition 12.6. If φ : C → D in C ∗ Cat is a unitary equivalence, then K C ∗ Cat ( φ ) : K C ∗ Cat ( C ) → K C ∗ Cat ( D ) is an equivalence. roof. We can consider the following commuting diagram of Z -graded abelian groups K ∗ ( C ) ∼ = (cid:47) (cid:47) K ∗ ( D ) π ∗ K C ∗ ( A f ( C )) ∼ = (cid:79) (cid:79) def π ∗ K C ∗ ( A f ( D )) ∼ = (cid:79) (cid:79) def π ∗ K C ∗ Cat ( C ) π ∗ K C ∗ Cat ( φ ) (cid:47) (cid:47) π ∗ K C ∗ Cat ( D ) (12.2)The K -theory groups in the upper line are defined in [Joa03, Def. 2.4] using the modulecategories of C and D . The upper horizontal morphism is an isomorphism by [Joa03,Lem. 2.7]. The vertical morphisms are isomorphisms by [Joa03, Thm. 3.11]. It followsthat the lower horizontal morphism is an isomorphism. Consequently,K C ∗ Cat ( φ ) : K C ∗ Cat ( C ) → K C ∗ Cat ( D )is an equivalence in Sp . Remark 12.7.
A proof of Proposition 12.6 which is independent of the results of [Joa03,Sec. 2] will be given in Corollary 16.5.We now use the functor A from (4.1). The universal property of A f together with (4.4)provides a natural transformation α : A f → A (12.3)of functors from C ∗ Cat nu i to C ∗ Alg nu , see, e.g., [BE20, Lem. 8.54].In order to provide a selfcontained presentation we give the proof of the following lemma.Let C be in C ∗ Cat nu . Lemma 12.8 ([BE20, Prop. 8.55]) . The morphism K C ∗ ( α C ) : K C ∗ ( A f ( C )) → K C ∗ ( A ( C )) (12.4) is an equivalence.Proof. If C is unital and has countable may objects, then the assertion of the propositionhas been shown by Joachim [Joa03, Prop. 3.8].First assume that C has countably many objects, but is possibly non-unital. Then thearguments from the proof of [Joa03, Prop. 3.8] are applicable and show that the canonicalmap α C : A f ( C ) → A ( C ) is a stable homotopy equivalence. Let use recall the constructionof the stable inverse β : A ( C ) → A f ( C ) ⊗ K , K := K ( H ) are the compact operators on the Hilbert space H := (cid:96) (Ob( C ) ∪ { e } ) , where e is an artificially added point. The assumption on the cardinality of Ob( C ) is madesince we want that K is the algebra of compact operators on a separable Hilbert space.Two points x, y in Ob( C ) ∪ { e } provide a rank-one operator Θ y,x in K ( H ) which sendsthe basis vector corresponding to x to the vector corresponding to y , and which vanisheson the orthogonal complement of x . The homomorphism β is given on A in Hom C ( x, y ) by β ( A ) := A ⊗ Θ y,x . If A and B are composable morphisms, then the relation Θ z,y Θ y,x = Θ z,x implies that β ( B ◦ A ) = β ( B ) β ( A ). Moreover, if A , B are not composable, then β ( B ) β ( A ) = 0. Finally, β ( A ) ∗ = β ( A ∗ ) since Θ ∗ y,x = Θ x,y . It follows that β is a well-defined ∗ -homomorphism.The argument now proceeds by showing that the composition ( α C ⊗ id K ( H ) ) ◦ β is homotopicto id A ( C ) ⊗ Θ e,e , and that the composition β ◦ α C is homotopic to id A f ( C ) ⊗ Θ e,e . Notethat in our setting C is not necessarily unital. In the following we directly refer to theproof of [Joa03, Prop. 3.8]. The only step in the proof of that proposition where theidentity morphisms are used is the definition of the maps denoted by u x ( t ) in the reference.But they in turn are only used to define the map denoted by Ξ later in that proof. Thecrucial observation is that we can define this map Ξ directly without using any identitymorphisms in C .We conclude that the canonical map K C ∗ ( α C ) : K ( A f ( C )) → K C ∗ ( A ( C )) is an equivalencefor C ∗ -categories C with countably many objects.In order to extend this to all C ∗ -categories, we use the fact that A f commutes withfiltered colimits which implies that A f ( C ) ∼ = colim C (cid:48) A f ( C (cid:48) ), where the colimit runs overthe filtered poset of all full subcategories with countably many objects. The connectingmaps of the indexing family of this colimit are functors which are injections on objects.We now argue that also A ( C ) ∼ = colim C (cid:48) A ( C (cid:48) ). Note that A is the composition of thefunctor A alg : C ∗ Cat nu → ∗ pre Alg nu C (see [Bun, Defn. 6.1]) with the completion functorCompl : ∗ pre Alg nu C → C ∗ Alg nu ([Bun, (3.17]). By construction the functor A alg preservesfiltered colimits with connecting maps which are injective on objects. The completionfunctor is a left-adjoint and therefore preserves all filtered colimits. This implies that A commutes with colim C (cid:48) .Since K C ∗ commutes with filtered colimits the morphismK C ∗ ( α C ) : K C ∗ ( A f ( C )) → K C ∗ ( A ( C ))is equivalent to the morphism colim C (cid:48) K C ∗ ( α C (cid:48) ) : colim C (cid:48) K C ∗ ( A f ( C (cid:48) )) → colim C (cid:48) K C ∗ ( A ( C (cid:48) )) . C (cid:48) appearing in the colimit have at most countably many objects wehave identified K C ∗ ( α C ) with a colimit of equivalences. Hence this morphism itself is anequivalence. Proposition 12.9.
The functor K C ∗ Cat sends excisive squares in C ∗ Cat nu to cocartesiansquares in Sp .Proof. Assume that we are given an excisive square (11.1). Using the exactness of thefunctor A shown in [Bun, Prop. 8.8.2] and 11.2.1 we get a commuting diagram in C ∗ Alg nu (cid:47) (cid:47) A ( A ) (cid:47) (cid:47) A ( B ) (cid:47) (cid:47) A ( B / A ) (cid:47) (cid:47) A f ( A ) (cid:15) (cid:15) (cid:47) (cid:47) α A (cid:79) (cid:79) A f ( B ) (cid:15) (cid:15) (cid:47) (cid:47) α B (cid:79) (cid:79) A f ( B / A ) (cid:15) (cid:15) α B / A (cid:79) (cid:79) A f ( C ) (cid:47) (cid:47) α C (cid:15) (cid:15) A f ( D ) (cid:47) (cid:47) α D (cid:15) (cid:15) A f ( D / C ) α D / C (cid:15) (cid:15) (cid:47) (cid:47) A ( C ) (cid:47) (cid:47) A ( D ) (cid:47) (cid:47) A ( D / C ) (cid:47) (cid:47) C ∗ and get adiagram · · · (cid:47) (cid:47) K C ∗ ( A ( A )) (cid:47) (cid:47) K C ∗ ( A ( B )) (cid:47) (cid:47) K C ∗ ( A ( B / A )) (cid:47) (cid:47) · · · K C ∗ Cat ( A ) (cid:39) (cid:79) (cid:79) (cid:47) (cid:47) (cid:15) (cid:15) K C ∗ Cat ( B ) (cid:15) (cid:15) (cid:39) (cid:79) (cid:79) (cid:47) (cid:47) (cid:15) (cid:15) K C ∗ Cat ( B / A ) (cid:39) ! (cid:15) (cid:15) (cid:39) (cid:79) (cid:79) K C ∗ Cat ( C ) (cid:39) (cid:15) (cid:15) (cid:47) (cid:47) K C ∗ Cat ( D ) (cid:39) (cid:15) (cid:15) (cid:47) (cid:47) K C ∗ Cat ( D / C ) (cid:39) (cid:15) (cid:15) · · · (cid:47) (cid:47) K C ∗ ( A ( C )) (cid:47) (cid:47) K C ∗ ( A ( D )) (cid:47) (cid:47) K C ∗ ( A ( D / C )) (cid:47) (cid:47) · · · (12.5)whose upper and lower horizontal lines are segments of fibre sequences by Proposition 12.1.3.The unmarked equivalences are instances of (12.4), while the morphism marked by ! is anequivalence by Proposition 12.6 and 11.2.3.We can now conclude that the left-middle square in (12.5) is cocartesian. Lemma 12.10.
For every set X we have K C ∗ Cat (0[ X ]) (cid:39) .Proof. We have A f (0[ X ]) ∼ = 0 and hence K C ∗ ( A f (0[ X ])) (cid:39) q ˜ C : ˜ C (cid:111) G → ˜ C (cid:111) r G is an isomorphism for any ˜ C in Fun ( BG, C ∗ Cat ) provided that G is amenable. If oneis interested in this isomorphism only after applying K -theory, then one can weaken theassumption on G from amenable to K -amenable. Since we consider discrete groups G wecan adopt the following definition: Definition 12.11 ([Cun83, Def. 2.2], [CCJ +
01, Sec. 1.3.2]) . The discrete group G is K -amenable if for every ˜ A in Fun ( BG, C ∗ Alg nu ) the morphism K C ∗ ( q ˜ A ) : K C ∗ ( ˜ A (cid:111) G ) → K C ∗ ( ˜ A (cid:111) r G ) is an equivalence. The class of K -amenable groups contains all amenable groups, but also all groups withthe Haagerup property (also often called a-T-menability), and hence for example also allCoxeter groups and all CAT(0)-cubical groups [CCJ +
01, Sec. 1.2].Let ˜ C be in Fun ( BG, C ∗ Cat ). Theorem 12.12. If G is K -amenable, then the morphism K C ∗ Cat ( q ˜ C ) : K C ∗ Cat ( ˜ C (cid:111) G ) → K C ∗ Cat ( ˜ C (cid:111) r G ) is an equivalence.Proof. We have the following commutative diagramK C ∗ Cat ( ˜ C (cid:111) G ) (12.4) ∼ = (cid:47) (cid:47) K C ∗ Cat ( q ˜ C ) (cid:15) (cid:15) K C ∗ ( A ( ˜ C (cid:111) G )) K C ∗ ( A ( q ˜ C )) (cid:15) (cid:15) K C ∗ ( A ( ˜ C ) (cid:111) G ) (10.20) ∼ = (cid:111) (cid:111) K C ∗ ( q A ( ˜ C ) ) (cid:15) (cid:15) K C ∗ Cat ( ˜ C (cid:111) r G ) (12.4) ∼ = (cid:47) (cid:47) K C ∗ ( A ( ˜ C (cid:111) r G )) K C ∗ ( A ( ˜ C ) (cid:111) r G ) (10.21) ∼ = (cid:111) (cid:111) where the right square is obtained by applying K C ∗ to the square(10.22). Because G is K -amenable, the right vertical arrow K C ∗ ( q A ( ˜ C ) ) is an equivalence. Therefore K C ∗ Cat ( q ˜ C )is an equivalence, too. K -theory of products of C ∗ -categories The main result of this section is Theorem 13.7 stating that the K -theory of a product ofadditive unital C ∗ -categories is equivalent to the product of the K -theories of the factors.82or finite products this immediately follows from Lemma 11.9 and Theorem 12.4. So theinteresting case are infinite families.In order to simplify the notation in this section we use the notation K ∗ ( A ) := π ∗ K C ∗ ( A )for the K -theory groups of a C ∗ -algebra A .Let A be an algebra and n, m in N . For a in A and i in { , . . . , n } and j in { , . . . , m } welet a [ i, j ] in Mat n,m ( A ) denote the matrix whose only non-zero entry is a in position ( i, j ).For i in { , . . . , n } we let (cid:15) A,n [ i ] : A → Mat n ( A ) (13.1)denote the injective (non-unital if n ≥
2) algebra homomorphism which sends a in A to a [ i, i ].Let A, B be ∗ -algebras. Recall from Definition 2.11 that an element u in B is a partialisometry if uu ∗ and u ∗ u are projections in B . Let h : A → B be a ∗ -homomorphism suchthat hu ∗ u = h . Then h (cid:48) := uhu ∗ : A → B is another ∗ -homomorphism.If A and B are C ∗ -algebras and the homomorphisms h, h (cid:48) : A → B are related as describedabove with u in the multiplier algebra of B , then we have an equality between the inducedmaps on K -theory groups h ∗ = h (cid:48)∗ : K ∗ ( A ) → K ∗ ( B ) , (13.2)see, e.g., [BE20, Rem. 8.44].If A is a C ∗ -algebra, n in N , and i in { , . . . , n } , then by the matrix stability of K C ∗ thehomomorphism of K -theory groups (cid:15) A,n [ i ] ∗ : K ∗ ( A ) → K ∗ ( Mat n ( A )) (13.3)induced by the homomorphism (13.1) of C ∗ -algebras is an isomorphism.We consider C in C ∗ Cat . If F is a finite subset of objects of C , then we have a unitalsubalgebra A ( F ) := (cid:77) C,C (cid:48) ∈ F Hom C ( C, C (cid:48) ) (13.4)of A alg ( C ), see (4.2) for notation. For n in N the inclusion A ( F ) → A ( C ) induces thehomomorphism of matrix algebras h F,n : Mat n ( A ( F )) → Mat n ( A ( C )) , where A ( C ) is as in (4.1). For an object C of C we use the notation (cid:96) C : End C ( C ) → A ( C ) (13.5)83or the canonical inclusion.Let C be in C ∗ Cat , F be a finite set of objects of C , and let n in N . Lemma 13.1.
Assume that C is additive. Then there is a partial isometry u in Mat n ( A ( C )) and an object C ( F, n ) in C such that h F,n u ∗ u = h F,n and h (cid:48) := uh F,n u ∗ has a factorization h (cid:48) : Mat n ( A ( F )) φ F,n −−→
End C ( C ( F, n )) (cid:96) C ( F,n ) −−−−→ A ( C ) (cid:15) A ( C ) ,n [1] −−−−−→ Mat n ( A ( C )) (13.6) where the isomorphism φ F,n will be constructed in the proof.Proof.
We consider the family ((
C, i )) C ∈ F,i ∈{ ...,n } of elements in F , i.e., every element of F is repeated n times. We then choose a sum (cid:0) C ( F, n ) , ( e ( C,i ) ) C ∈ F,i ∈{ ,...,n } (cid:1) (13.7)of this finite family, see Definition 3.2.We can view morphisms in C as elements of A ( C ) in a canonical way. A morphism betweenobjects in F is an element of A ( F ). We have an isomorphism φ F,n : Mat n ( A ( F )) → End C ( C ( F, n )) , (13.8)that sends the matrix f [ i, i (cid:48) ] with f : C (cid:48) → C in Mat n ( A ( F )) to e C,i f e ∗ C (cid:48) ,i (cid:48) in End C ( C ( F, n )).One checks that (cid:15) A ( C ) ,n [1] ◦ (cid:96) C ( F,n ) ◦ φ F,n ( − ) = n (cid:88) i,i (cid:48) =1 (cid:88) C,C (cid:48) ∈ F e C,i [1 , i ]( − ) e ∗ C (cid:48) ,i (cid:48) [ i,
1] (13.9)as maps
Mat n ( A ( F )) → Mat n ( A ( C )). We define a matrix in Mat n ( A ( C )) by u := n (cid:88) i =1 (cid:88) C ∈ F e C,i [1 , i ] . (13.10)Using the orthogonality relations for the family ( e C,i ) C ∈ F,i ∈{ ,...,n } considered as elementsin A ( C ) we calculate that uu ∗ = id C ( F,n ) [1 , , u ∗ u = 1 Mat n ( A ( F )) . (13.11)The second equation in (13.11) immediately implies that h F,n = h F,n u ∗ u . We now calculate h (cid:48) := uh F,n u ∗ = n (cid:88) i,i (cid:48) =1 (cid:88) C,C (cid:48) ∈ F e C,i [1 , i ] h F,n e ∗ C (cid:48) ,i (cid:48) [ i (cid:48) , (13.9) = (cid:15) A ( C ) ,n [1] ◦ (cid:96) C ( F,n ) ◦ φ F,n . emark 13.2. In this remark we recall the standard way to present elements in K ( A )for a C ∗ -algebra A , see e.g. [Bla98].Let A + denote the unitalization of A . If P, ˜ P is a pair of projections in Mat n ( A + ) suchthat P ≡ ˜ P modulo Mat n ( A ), then we have a K -theory class [ P, ˜ P ] in K ( A ). Every classin K ( A ) can be represented in this way.We let [ P, ˜ P ] n be the class represented by this pair of projections in K ( Mat n ( A )). Thenusing the isomorphism (13.3) we have the equality[ P, ˜ P ] = (cid:15) A,n [1] − ∗ [ P, ˜ P ] n . (13.12)If A is unital and P is a projection in A , then we get a class [ P ] in K ( A ).If [ P, ˜ P ] = 0, then after increasing n if necessary, there exists a partial isometry U in Mat n ( A + ) such that U U ∗ = P and U ∗ U = ˜ P .Let C be in C ∗ Cat . Lemma 13.3.
We assume that C is additive.1. For every class p in K ( A ( C )) there exists an object C and projections P, ˜ P in End C ( C ) such that (cid:96) C, ∗ ([ P ] − [ ˜ P ]) = p .2. If P, ˜ P in End C ( C ) are projections such that (cid:96) C, ∗ ([ P ] − [ ˜ P ]) = 0 , then there exists apartial isometry U in End C ( C ) such that U U ∗ = P and U ∗ U = ˜ P .Proof. Let p be a class in K ( A ( C )). Then there exists an n in N and a pair of projections P (cid:48) , ˜ P (cid:48) in Mat n ( A ( C ) + ) such that P (cid:48) ≡ ˜ P (cid:48) modulo Mat n ( A ( C )) and p = [ P (cid:48) , ˜ P (cid:48) ].We first note that the dense subalgebra A alg ( C ) + of A ( C ) + is closed under holomorphicfunctional calculus. Every element of A alg ( C ) + is contained in A ( F ) + for a sufficiently largefinite set of objects of C . The same applies to n -by- n matrices. We can therefore modifythe choices of P (cid:48) and ˜ P (cid:48) such that P (cid:48) , ˜ P (cid:48) belong to Mat n ( A ( F ) + ) for a sufficiently largeset F of objects of C . We write [ P (cid:48) , ˜ P (cid:48) ] F for the corresponding class in K ( Mat n ( A ( F ))).Since A ( F ) is unital we have decompositions A ( F ) + ∼ = A ( F ) ⊕ C , Mat n ( A ( F ) + ) ∼ = Mat n ( A ( F )) ⊕ Mat n ( C ) . If we take the components P (cid:48)(cid:48) , ˜ P (cid:48)(cid:48) of the projections P (cid:48) , ˜ P (cid:48) in Mat n ( A ( F )), then we havethe equality [ P (cid:48) , ˜ P (cid:48) ] F = [ P (cid:48)(cid:48) ] − [ ˜ P (cid:48)(cid:48) ]85n K ( Mat n ( A ( F ))).Using the notation introduced in Lemma 13.1 we set C := C ( F, n ), P := φ F,n ( P (cid:48)(cid:48) ) and˜ P := φ F,n ( ˜ P (cid:48)(cid:48) ). We have the chain of equalities p = [ P (cid:48) , ˜ P (cid:48) ] (13.12) = (cid:15) A ( C ) ,n [1] − ∗ [ P (cid:48) , ˜ P (cid:48) ] n = (cid:15) A ( C ) ,n [1] − ∗ h F,n, ∗ [ P (cid:48) , ˜ P (cid:48) ] F = (cid:15) A ( C ) ,n [1] − ∗ h F,n, ∗ ([ P (cid:48)(cid:48) ] − [ ˜ P (cid:48)(cid:48) ]) (13.2) = (cid:15) A ( C ) ,n [1] − ∗ h (cid:48)∗ ([ P (cid:48)(cid:48) ] − [ ˜ P (cid:48)(cid:48) ]) (13.6) = (cid:96) C ( F,n ) , ∗ ([ φ F,n ( P (cid:48)(cid:48) )] − [ φ F,n ( ˜ P (cid:48)(cid:48) )])= (cid:96) C, ∗ ([ P ] − [ ˜ P ]) . This finishes the verification of Assertion 1.We now show the Assertion 2. For n in N we set P (cid:48) := (cid:96) C ( P )[1 ,
1] and ˜ P (cid:48) := (cid:96) C ( P )[1 ,
1] in
Mat n ( A ( C ) + ). By assumption we can choose n and a partial isometry U (cid:48) in Mat n ( A ( C ) + )such that U (cid:48) U (cid:48) , ∗ = P (cid:48) and U (cid:48) , ∗ U (cid:48) = ˜ P (cid:48) .Note that P (cid:48) and ˜ P (cid:48) belong to the subalgebra Mat n ( A ( C )). This implies that U (cid:48) belongsto Mat n ( A ( C )).Let U (cid:48)(cid:48) := P (cid:48) U (cid:48) ˜ P (cid:48) . Then we calculate in a straightforward manner that U (cid:48)(cid:48) U (cid:48)(cid:48) , ∗ = P (cid:48) , U (cid:48)(cid:48) , ∗ U (cid:48)(cid:48) = ˜ P (cid:48) . We furthermore observe that U (cid:48)(cid:48) = (cid:96) C ( U )[1 ,
1] for a uniquely determined partial isometry U in End C ( C ) which satisfies U U ∗ = P and U ∗ U = ˜ P .Let A be a unital C ∗ -algebra, U be a unitary in A , and V : [0 , → Mat n ( A ) be a Lipschitzcontinuous path of unitaries from ( U − A )[1 ,
1] + 1
A,n to 1
A,n .The following lemma is inspired by [WY20, Proof of 12.6.3]. It improves the Lipschitzconstant of the path to 7 π at the cost of increasing the size of matrizes. Lemma 13.4.
There exists n (cid:48) in N and a π -Lipschitz continuous path V (cid:48) : [0 , → Mat n (cid:48) ( A ) of unitaries from ( U − A )[1 ,
1] + 1
A,n (cid:48) to A,n (cid:48) .Proof.
Assume that V : [0 , → Mat n ( A ) is a Lipschitz continuous path of unitaries from( U − A )[1 ,
1] + 1
A,n to 1
A,n with Lipschitz constant bounded by C . Then we will construct86 new path V (cid:48) : [0 , → Mat n ( A ) of unitaries with Lipschitz constant bounded by π + C from ( U − A )[1 ,
1] + 1 A, n to 1 A, n . To this end we write( U − A )[1 ,
1] + 1 A, n = V (0) 0 00 V (1 /
2) 00 0 V (1) V (1 / ∗
00 0 V (1) ∗ . We have a path defined on [0 , / V (1 / − t/ ∗
00 0 V (1 − t/ ∗ from V (1 / ∗
00 0 V (1) ∗ to V (0) ∗
00 0 V (1 / ∗ . This path has Lipschitz constant 3 / C . We furthermore have a rotation path defined on[2 / ,
1] of speed 3 π/ A V (0) ∗
00 0 V (1 / ∗ to V (0) ∗ V (1 / ∗
00 0 1 A . The product of the concatenation of these paths with V (0) 0 00 V (1 /
2) 00 0 V (1 A ) is a path from ( U − A )[1 ,
1] + 1 A, n to 1 A, n with Lipschitz constant bounded by π + C .The fixed point of the iteration C ⇒ π C π .By iterating the construction above sufficiently often we can produce a path as asserted. Remark 13.5.
In this remark we recall the standard way to represent elements in K ( A )for a C ∗ -algebra A , see e.g. [Bla98].A unitary U in Mat n ( A + ) with U ≡ n modulo Mat n ( A ) represents a class [ U ] in K ( A ).Every class in K ( A ) can be represented in this way.We let [ U ] n denote the class of U in K ( Mat n ( A )). Then using the isomorphism (13.3) wehave the equality [ U ] = (cid:15) A,n [1] − ∗ [ U ] n . (13.13)87f A is unital, then a unitary U as above is of the form ( U (cid:48) − A,n , n ) for a unitary U (cid:48) in Mat n ( A ). If U (cid:48) is a unitary in Mat n ( A ), then we set [ U (cid:48) ] := [( U (cid:48) − A,n , n )].Assume that U and ˜ U are two such unitaries and that [ U ] = [ U (cid:48) ]. Then, after increasing n if necessary, there exists a path V : [0 , → Mat n ( A + ) of unitaries from U to ˜ U such that V ( t ) ≡ n for all t in [0 , A is unital, then the path is of the form V = ( V (cid:48) − A,n , n ),where V (cid:48) is a path of unitaries in Mat n ( A ) from U (cid:48) to ˜ U (cid:48) .Let C be in C ∗ Cat . Lemma 13.6.
We assume that C is additive.1. For every class u in K ( A ( C )) there exists an object C and a unitary U in End C ( C ) such that u = (cid:96) C, ∗ [ U ] .2. Assume that U in End C ( C ) is a unitary such that (cid:96) C, ∗ [ U ] = 0 . Then there existsan object C (cid:48) , an isometry u : C → C (cid:48) , and a π -Lischitz path V : [0 , → End C ( C (cid:48) ) from uU u ∗ + ( id C (cid:48) − uu ∗ ) to id C (cid:48) .Proof. Let u be a class u in K ( A ( C )). Then there exists n in N and a unitary U (cid:48) in Mat n ( A ( C ) + ) such that U (cid:48) ≡ n modulo Mat n ( A ( C )) and [ U (cid:48) ] = u . As in the proof ofLemma 13.3 we can modify U (cid:48) such that it belongs to Mat n ( A ( F ) + ) for a sufficiently largeset F of objects of C . Since A ( F ) is unital we obtain a unitary U (cid:48)(cid:48) in Mat n ( A ( F )) such that U (cid:48) = ( U (cid:48)(cid:48) − A ( F ) ,n , n ). We let [ U (cid:48)(cid:48) ] denote the corresponding class in K ( Mat n ( A ( F ))).We set C := C ( F, n ) and define the unitary U := φ F,n ( U (cid:48)(cid:48) ) in End C ( C ), where C ( F, n ) isas in (13.7) and φ F,n is as in (13.8). We have the following chain of equalities u = [ U (cid:48) ] (13.13) = (cid:15) A ( C ) ,n [1] − ∗ [ U (cid:48) ] n = (cid:15) A ( C ) ,n [1] − ∗ h F,n, ∗ [ U (cid:48)(cid:48) ] (13.2) = (cid:15) A ( C ) ,n [1] − ∗ h (cid:48)∗ [ U (cid:48)(cid:48) ] (13.6) = (cid:96) C ( F,n ) , ∗ [ φ F,n ( U (cid:48)(cid:48) )]= (cid:96) C, ∗ [ U ]This finishes the proof of Assertion 1.We now show Assertion 2. Since (cid:96) C, ∗ [ U ] = 0 there exists n in N and a path of unitaries V (cid:48) : [0 , → Mat n ( A ( C ) + ) from (( U − C )[1 , , n ) to 1 n such that V (cid:48) ( t ) ≡ n for all t in[0 , Mat n ( A ( F ) + ) for a sufficientlylarge set of objects F containing C . Since A ( F ) is unital we can write V (cid:48) := ( V (cid:48)(cid:48) − A ( F ) ,n , n )for a path V (cid:48)(cid:48) of unitaries in Mat n ( A ( F )) from ( U − id C )[1 ,
1] + 1 A ( F ) ,n to 1 A ( F ) ,n .88e now apply Lemma 13.4. It provides a 7 π -Lipschitz path V (cid:48)(cid:48)(cid:48) : [0 , → Mat n (cid:48) ( A ( F )) ofunitaries from ( U − id C )[1 ,
1] + 1 A ( F ) ,n (cid:48) to 1 A ( F ) ,n (cid:48) .We now consider object C (cid:48) := C ( F, n (cid:48) ) (see (13.7)) and the isometry u := e C, : C → C (cid:48) .We furthermore define the 7 π -Lipschitz path V := φ F,n (cid:48) ( V (cid:48)(cid:48)(cid:48) ), where φ F,n (cid:48) is as in (13.8).This path does the job since V (0) = φ F,n (cid:48) ( V (cid:48)(cid:48)(cid:48) (0)) = φ F,n (cid:48) (( U − id C )[1 ,
1] + 1 A ( F ) ,n (cid:48) ) = uU u ∗ + ( id C (cid:48) − uu ∗ )and V (1) = φ F,n (cid:48) ( V (cid:48)(cid:48)(cid:48) (0)) = φ F,n (cid:48) (1 A ( F ) ,n (cid:48) ) = id C (cid:48) . Let ( C i ) i ∈ I be a family in C ∗ Cat . For every i in I the projection p i : (cid:81) i ∈ I C i → C i induces a morphism of spectraK C ∗ Cat ( p i ) : K ( (cid:89) i ∈ I C i ) → K C ∗ Cat ( C i ) . Theorem 13.7. If C i is additive for every i in I , then the morphism of spectra K C ∗ Cat (cid:0) (cid:89) i ∈ I C i (cid:1) → (cid:89) i ∈ I K C ∗ Cat ( C i ) (13.14) induced by the family ( K ( p i )) i ∈ I is an equivalence.Proof. We consider the diagram A f ( (cid:81) i ∈ I C i ) (cid:47) (cid:47) (cid:15) (cid:15) (cid:81) i ∈ I A f ( C i ) (cid:15) (cid:15) A ( (cid:81) i ∈ I C i ) (cid:81) i ∈ I A ( C i ) (13.15)in C ∗ Alg nu , where left upper horizontal morphism is induced by the family ( A f ( p i )) i ∈ I .The vertical maps are instances of (12.3) and induce isomorphisms in K -theory groups byLemma 12.8. Hence we get a square K ∗ ( (cid:81) i ∈ I C i ) (13.14) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) (cid:81) i ∈ I K ∗ ( C i ) ∼ = (cid:15) (cid:15) K ∗ ( A f ( (cid:81) i ∈ I C i )) ! (cid:47) (cid:47) ∼ = (cid:15) (cid:15) K ∗ ( (cid:81) i ∈ I A f ( C i )) !! (cid:47) (cid:47) (cid:81) i ∈ I K ∗ ( A f ( C i )) ∼ = (cid:15) (cid:15) K ∗ ( A ( (cid:81) i ∈ I C i )) ? (cid:47) (cid:47) (cid:81) i ∈ I K ∗ ( A ( C i )) (13.16)where the homomorphism marked by ! is induced from the horizontal homomorphism in(13.15), and the homomorphism !! is the canonical comparison homomorphism. The upper89ertical isomorphisms reflect Definition 12.3, while the lower vertical isomorphisms areinstances of (12.4). In order to show that (13.14) is an isomorphism it suffices to showthat the morphism ? (defined as the up-right-down composition) is an isomorphism. Inview of Bott periodicity (Remark 12.2.4) is suffices to consider the cases ∗ = 0 and ∗ = 1.In the following argument we will frequently use the following fact. Let C be in C ∗ Cat nu and C be an object of C . Then we have a commuting triangle End C ( C ) (cid:96) fC (cid:121) (cid:121) (cid:96) C (cid:37) (cid:37) A f ( C ) (12.4) (cid:47) (cid:47) A ( C ) (13.17)where both diagonal morphisms are inclusions of closed subalgebras.surjectivity of ? in (13.16) for ∗ = 0:Let ( p i ) i ∈ I be a class in (cid:81) i ∈ I K ( A ( C i )). By Lemma 13.3.1 for every i in I there exists anobject C i in C i and projections P i , ˜ P i in End C i ( C i ) such that p i = (cid:96) C i , ∗ ([ P i ] − [ ˜ P i ]) . We can form projections ( P i ) i ∈ I , ( ˜ P i ) i ∈ I in End (cid:81) i ∈ I C i (( C i ) i ∈ I ). Using (13.17) we see thatthe class (cid:96) ( C i ) i ∈ I , ∗ ([( P i ) i ∈ I ] − [( ˜ P i ) i ∈ I ])in K ( A ( (cid:81) i ∈ I C i )) provides a preimage of the class ( p i ) i ∈ I under the morphism ?.injectivity of ? in (13.16) for ∗ = 0:We note that the product category (cid:81) i ∈ I C i is again additive. Indeed, we can form sumscomponentwise (see Lemma 7.4). Let p be a class in K ( A ( (cid:81) i ∈ I C i )) which is sent tozero by ?. By Lemma 13.3.1 there is an object ( C i ) i ∈ I of (cid:81) i ∈ I C i and projections P, ˜ P in End (cid:81) i ∈ I C i (( C i ) i ∈ I ) such that (cid:96) ( C i ) i ∈ I , ∗ ([ P ] − [ ˜ P ]) = p . We have P = ( P i ) i ∈ I and ˜ P = ( ˜ P i ) i ∈ I for projections P i , ˜ P i in End C i ( C i ). By assumptionon p and (13.17) for every i in I we have (cid:96) C i , ∗ ([ P i ] − [ ˜ P i ]) = 0. By Lemma 13.3.2 for every i in I exists a partial isomety U i in End C i ( C i ) such that U i U ∗ i = P i and U ∗ i U i = ˜ P i . Then U := ( U i ) i ∈ I is a partial isometry in End (cid:81) i ∈ I C i (( C i ) i ∈ I ) such that U U ∗ = P and U ∗ U = ˜ P .Then [ P ] − [ ˜ P ] = 0 in K ( End (cid:81) i ∈ I C i (( C i ) i ∈ I )) and therefore p = (cid:96) ( C i ) i ∈ I , ∗ ([ P ] − [ ˜ P ]) = 0.surjectivity of ? in (13.16) for ∗ = 1:Let ( u i ) i ∈ I be a class in (cid:81) i ∈ I K ( A ( C i )). By Lemma 13.6.1 for every i in I there exists anobject C i in C i and a unitary U i in End C i ( C i ) such that (cid:96) C i , ∗ [ U i ] = u i . The family ( U i ) i ∈ I
90s a unitary in
End (cid:81) i ∈ I C i (( C i ) i ∈ I ). Using (13.17) we see that the class (cid:96) ( C i ) i ∈ I , ∗ [( U i ) i ∈ I ] in K ( A ( (cid:81) i ∈ I C i )) is the desired preimage of the class ( u i ) i ∈ I under ?.injectivity of ? in (13.16) for ∗ = 1:Let u be a class in K ( A ( (cid:81) i ∈ I C i )) which is sent to zero by ?. By Lemma 13.6.1 there is anobject C := ( C i ) i ∈ I in (cid:81) i ∈ I C i and a unitary U in End (cid:81) i ∈ I C i ( C ) such that (cid:96) ( C i ) i ∈ I , ∗ [ U ] = u .We have U = ( U i ) i ∈ I for unitaries U i in End C i ( C i ). By assumption on u and (13.17) wehave (cid:96) C i , ∗ [ U i ] = 0 for all i in I . By Lemma 13.6.2 for every i we can find an object C (cid:48) i in C i , an isometry u i : C i → C (cid:48) i , and a 7 π -Lipschitz path V i : [0 , → End C i ( C (cid:48) i ) from u i U i u ∗ i + ( id C (cid:48) i − u i u ∗ i ) to id C (cid:48) i . We define the object C (cid:48) := ( C (cid:48) i ) ∈ I in (cid:81) i ∈ I C i and theisometry u := ( u i ) i ∈ I : C → C (cid:48) in (cid:81) i ∈ I C i . Then V := ( V i ) i ∈ I is a path in End (cid:81) i ∈ I C i ( C (cid:48) )from uU u ∗ + ( id C (cid:48) − uu ∗ ) to id C (cid:48) . At this point, in order to see that V is continuousone needs the uniform bound on the Lipshitz constants of the paths V i . This shows that[ uU u ∗ + ( id C (cid:48) − uu ∗ )] = 0 in K ( End (cid:81) i ∈ I C i ( C (cid:48) )). We have (cid:96) C = (cid:96) C u ∗ u in A ( C ) and thefactorization u(cid:96) C u ∗ : End (cid:81) i ∈ I C i ( C ) φ −→ End (cid:81) i ∈ I C i ( C (cid:48) ) (cid:96) C (cid:48) −−→ A ( (cid:89) i ∈ I C i ) , (13.18)where φ ( − ) := u ( − ) u ∗ . Note that these homomorphisms are not unital. To apply thesemaps to unitaries representing K -theory classes we must extend them to the unitalizations.This leads to the formula φ ∗ [ U ] = [ φ ( U ) + ( id C (cid:48) − φ ( id C ))] = [ uU u ∗ + ( id C (cid:48) − uu ∗ )] . The homotopy V whitnesses the fact that φ ∗ [ U ] = 0. Finally, we have u = (cid:96) ( C i ) i ∈ I , ∗ [ U ] (13.2) = u(cid:96) ( C i ) i ∈ I u ∗ [ U ] (13.18) = (cid:96) C (cid:48) , ∗ φ ∗ [ U ]= 0 .
14 Morita invariance
In this section we recall the notion of a Morita equivalence between C ∗ -categories. We thenshow that the reduced crossed product preserves Morita equivalences. We then considerMorita invariant homological functors and verify that K C ∗ Cat is an example of such afunctor.Recall from Definition 3.5 that E in C ∗ Cat is called additive if it admits orthogonal sumsfor all finite families of objects. Let i : D → E be a functor in C ∗ Cat . Definition 14.1.
The functor i presents E as the additive completion of D if: . The functor i is fully faithful.2. The C ∗ -category E is additive.3. Every object of E is unitarily isomorphic to a finite orthogonal sum of objects in theimage of i . Example 14.2. If X is a set, then the functor ∅ → X ] presents 0[ X ] as an additivecompletion of ∅ . Remark 14.3.
Let C ∗ Cat ⊕ be the full subcategory of C ∗ Cat of additive C ∗ -categories.Then there exists a functor and a natural transformation( − ) ⊕ : C ∗ Cat → C ∗ Cat ⊕ , id → ( − ) ⊕ , such that for every C in C ∗ Cat the morphism C → C ⊕ presents C ⊕ as the additivecompletion of C , see [DL98, Sec. 2] or [DT14, Defn. 2.8].If one passes to ∞ -categories, then this additive completion functor fits into an adjunction.Let W be the set of unitary equivalences in C ∗ Cat . Then ( − ) ⊕ descends to the left-adjointof an adjunction (see [DT14, Lem. 2.12] for a 2-categorial formulation)( − ) ⊕ : C ∗ Cat [ W − ] (cid:28) C ∗ Cat [ W − ] ⊕ : incl , where C ∗ Cat [ W − ] ⊕ is the full subcategory of C ∗ Cat [ W − ] of additive C ∗ -categories.The details can be understood similarly as in the case of additive categories [BEKW20b,Cor. 2.62], using a Bousfield localization of model category structures as constructed in[DT14]. See also [Bun19] and Remark 8.6.Recall from Definition 2.15 that E in C ∗ Cat is idempotent complete if every projection in E is effective. We again consider a functor i : D → E in C ∗ Cat . Definition 14.4.
The functor i presents E as the idempotent completion of D if:1. The functor i is fully faithful.2. The C ∗ -category E is idempotent complete.3. For every object E in E there is some object D in D and an isometry u : E → i ( D ) . Remark 14.5.
For simplicity we consider the idempotent completion only of additive C ∗ -categories. Let C ∗ Cat
Idem ⊕ denote the full subcategory of Cat ⊕ of idempotent complete,additive C ∗ -categories. Then there exists a functor and a natural transformationIdem : C ∗ Cat ⊕ → C ∗ Cat
Idem ⊕ , id → Idem , C in C ∗ Cat ⊕ the functor C → Idem( C ) presents Idem( C ) as theidempotent completion of C , see [DT14, Defn. 2.15].Note that by [DT14, Rem. 2.19] the idempotent completion of an additive C ∗ -categoryis again additive. Furthermore, in the same remark it is explained that the operationsof forming additive completions and of idempotent completions do not commute sincethe additive completion of an idempotent complete C ∗ -category does not need to beidempotent complete.The idempotent completion functor descends to an adjunction between ∞ -categories (see[DT14, Defn. 2.17] for a 2-categorical formulation)Idem : C ∗ Cat [ W − ] ⊕ (cid:28) C ∗ Cat [ W − ] Idem ⊕ : incl , where C ∗ Cat [ W − ] Idem ⊕ is the full subcategory of C ∗ Cat [ W − ] ⊕ of idempotent complete(and additive) C ∗ -categories. The details are again similar to the case of additive categories[BEKW20b, Cor. 3.7], again using a Bousfield localization of model category structuresconstructed in [DT14].By composing the additive and idempotent completion functors and the correspondingnatural transformations we obtain a functor and a natural transformation( − ) (cid:93) := Idem ◦ ( − ) ⊕ : C ∗ Cat → C ∗ Cat
Idem ⊕ , id → ( − ) (cid:93) . (14.1)For every C in C ∗ Cat the functor C → C (cid:93) is injective on objects by construction. Definition 14.6 ([DT14, Defn. 4.4]) . We define W Morita to be the set of functors in C ∗ Cat which are sent to unitary equivalences by ( − ) (cid:93) . The functors in W Morita are called Moritaequivalences.
Note that we have an inclusion W ⊆ W Morita .We consider the Dwyer–Kan localization (cid:96)
Morita : C ∗ Cat → C ∗ Cat [ W − ] . (14.2)The ∞ -category C ∗ Cat [ W − ] can be modeled by a cofibrantly generated simplicialmodel category structure on C ∗ Cat [DT14, Thm. 4.9]. There is a Bousfield localization L Morita : C ∗ Cat [ W − ] (cid:28) C ∗ Cat [ W − ] . Example 14.7.
Let ˜ A be a unital C ∗ -algebra with a G -action by automorphism. We canthen consider the C ∗ -category Hilb G ( ˜ A ) as Example 10.3. It contains the subcategory Hilb G ( ˜ A ) fg , proj of finitely generated, projective Hilbert A -modules, and we may consider93he object ˜ A in Hilb G ( ˜ A ) fg , proj as a C ∗ -category with a single object. The inclusion˜ A → Hilb G ( ˜ A ) fg , proj is a Morita equivalence: in order to see this we consider the chain˜ A → Hilb G ( ˜ A ) fg , free → Hilb G ( ˜ A ) fg , proj . The first functor presents
Hilb G ( ˜ A ) fg , free as the additive completion of ˜ A , and the secondfunctor presents Hilb G ( ˜ A ) fg , proj as the idempotent completion of Hilb G ( ˜ A ) fg , free .Our next goal is to show that the reduced crossed product preserves Morita equivalences.Let ˜ D → ˜ E be a morphism in Fun ( BG, C ∗ Cat ). It is called a Morita equivalence if theinduced morphism D → E between the underlying C ∗ -categories is a Morita equivalence.We now consider a chain of inclusions of full G -invariant subcategories˜ D → ˜ E → ˜ C in Fun ( BG, C ∗ Cat ). We assume that ˜ C is | G | -additive and idempotent complete. Proposition 14.8. If ˜ D → ˜ E is a Morita equivalence, then ˜ D (cid:111) C r G → ˜ E (cid:111) C r G is aMorita equivalence.Proof. The case of the trivial group G is obvious. Thus we assume that | G | ≥
2. Then ˜ C is additive.By abuse of notation we let D ⊕ and E ⊕ denote the G -invariant full subcategories of allobjects of C which are unitarily isomorphic to orthogonal sums of finite families of objectsof D or E , respectively. The inclusions D → D ⊕ and E → E ⊕ present their targets asadditive completions of their respective domains.By further abuse of notation we let Idem( D ⊕ ) and Idem( E ⊕ ) denote the full subcategoryof objects in C which are unitarily isomorphic to images of projections on objects of D ⊕ or E ⊕ , respectively. The inclusions D ⊕ → Idem( D ⊕ ) and E ⊕ → Idem( E ⊕ ) present theirtargets as the idempotent completions of their domains. We have a commuting diagram D (cid:47) (cid:47) (cid:15) (cid:15) D ⊕ (cid:15) (cid:15) (cid:47) (cid:47) Idem( D ⊕ ) E (cid:47) (cid:47) E ⊕ (cid:47) (cid:47) Idem( E ⊕ )of G -invariant subcategories of C . The assumption that the left vertical morphism is aMorita equivalence implies that the right vertical morphism is an equality.Let ( C i ) i ∈ I be a finite family of objects in ˜ C (cid:111) r G and ( C, ( e i ) i ∈ I ) a representative ofits orthogonal sum in C . Then ( C, ( e i , e ) i ∈ I ) (see [Bun, Defn. 5.1] for the notation for94orphisms in crossed products) represents its orthogonal sum in ˜ C (cid:111) r G . We call thisrepresentative a standard representative. We can conclude that ˜ C (cid:111) r G is additive.We furthermore see that we have an inclusion˜ E ⊕ (cid:111) r G → ( ˜ E (cid:111) r G ) ⊕ which is a unitary equivalence. The target may have more objects since we have more mor-phisms at hand to present them as orthogonal sums. But these additional representativesof orthogonal sums are unitarily isomorphic to standard representatives.We let ˜ C (cid:111) r G → Idem( ˜ C (cid:111) r G )denote the idempotent completion of ˜ C (cid:111) r G . This functor is fully faithful and injectiveon objects. We obtain a square˜ D (cid:111) r G (cid:47) (cid:47) (cid:15) (cid:15) Idem(( ˜ D (cid:111) r G ) ⊕ ) (cid:15) (cid:15) ˜ E (cid:111) r G (cid:47) (cid:47) Idem(( ˜ E (cid:111) r G ) ⊕ )of inclusions of full subcategories of Idem( ˜ C (cid:111) r G ). We must show that the right verticalmap is an equality. It suffices to show that it is surjective on objects.Let C be an object of Idem(( ˜ E (cid:111) r G ) ⊕ ). We must show that C belongs to Idem(( ˜ D (cid:111) r G ) ⊕ ).Since all objects in ( ˜ E (cid:111) r G ) ⊕ are unitarily isomorphic to standard representatives, thereexists an object E in E ⊕ and an isometry ˜ u : C → E in ˜ C (cid:111) r G . It suffices to find an object D of D ⊕ and an isometry ˜ v : C → D in ˜ C (cid:111) r G . Since D → E is a Morita equivalencethere exists an object D of D ⊕ and an isometry w : E → D in C . Then the composition v : C ˜ u −→ E ( w,e ) −−−→ D is the desired isometry in ˜ C (cid:111) r G .Let Hg : C ∗ Cat → M be a functor with values in some ∞ -category. Definition 14.9. Hg is Morita invariant if it sends the morphisms in W Morita to equiva-lences.
Let Hg : C ∗ Cat → M be a functor. Lemma 14.10.
The following are equivalent: . Hg is Morita invariant.2. Hg sends the following functors in C ∗ Cat to equivalences:a) Unitary equivalences.b) Fully faithful functors i : D → E satisfying:i. i is injective on objects.ii. E is additive and idempotent complete.iii. For every object E in E there is a finite family ( D k ) k ∈ K of objects in D and an isometry E → (cid:76) k ∈ K i ( D k ) .Proof. (1) ⇒ (2): Unitary equivalences and functors i as in 2b are Morita equivalences.If Hg is Morita invariant, then it sends these functors to equivalences.(2) ⇒ (1): Let C → D be a Morita equivalence. We must show that Hg( C ) → Hg( D ) isan equivalence.We have a commuting square C (cid:47) (cid:47) (cid:15) (cid:15) D (cid:15) (cid:15) C (cid:93) (cid:39) (cid:47) (cid:47) D (cid:93) Since the vertical morphisms are fully faithful we conclude that C → D is fully faithful.We therefore have a factorization C → C (cid:48) → D , where C (cid:48) is a full subcategory of D givenby the image of the functor C → D . Then C → C (cid:48) is a unitary equivalence and C (cid:48) → D a Morita equivalence. Since Hg( C ) → Hg( C (cid:48) ) is an equivalence it remains to show thatHg( C (cid:48) ) → Hg( D ) is an equivalence. To this end we consider the commuting squareHg( C (cid:48) ) (cid:47) (cid:47) (cid:15) (cid:15) Hg( D ) (cid:15) (cid:15) Hg( C (cid:48) ,(cid:93) ) (cid:39) (cid:47) (cid:47) Hg( D (cid:93) )The functors C (cid:48) → C (cid:48) ,(cid:93) and D → D (cid:93) satisfy the conditions in 2b. Hence the verticalmorphisms are equivalences by assumption on Hg. We conclude that the upper horizontalmorphism is an equivalence.Let Hg : C ∗ Cat → M be a homological functor. By definition it sends zero categoriesto zero, and it preserves some finite products by Lemma 11.9. But it is not clear thatit preserves finite coproducts, or how to characterize the value H ( ∅ ). Morita invarianceimproves the situation. 96 emma 14.11. If Hg is Morita invariant, then it preserves finite coproducts.Proof. Let ( C i ) i ∈ I be a finite family in C ∗ Cat . We must show that the canonical map (cid:97) i ∈ I Hg( C i ) → Hg (cid:0) (cid:97) i ∈ I C i (cid:1) (14.3)is an equivalence.We consider the Dwyer–Kan localization (cid:96) Morita : C ∗ Cat → C ∗ Cat [ W − ] from (14.2).By the universal property of the Dwyer–Kan localization the functor Hg has an essentiallyunique factorization C ∗ Cat Hg (cid:47) (cid:47) (cid:96) Morita (cid:40) (cid:40) M .C ∗ Cat [ W − ] Hg Morita (cid:55) (cid:55) (14.4)We use the factorization (14.4) in order to factorize (14.3) as (cid:97) i ∈ I Hg Morita ( (cid:96) Morita ( C i )) → Hg Morita (cid:0) (cid:97) i ∈ I (cid:96) Morita ( C i ) (cid:1) → Hg Morita (cid:0) (cid:96)
Morita (cid:0) (cid:97) i ∈ I C i (cid:1)(cid:1) . (14.5)The ∞ -category C ∗ Cat [ W − ] can be modeled by a cofibrantly generated simplicialmodel category structure on C ∗ Cat [DT14, Thm. 4.9] in which every object is cofibrant. Bythe latter property the canonical map (cid:96) i ∈ I (cid:96) Morita ( C i ) → (cid:96) Morita ( (cid:96) i ∈ I C i ) is an equivalencein C ∗ Cat [ W − ]. Hence the second morphism in (14.5) is an equivalence. We now claimthat Hg Morita preserves finite products.For the moment assume the claim. Since M (as a stable ∞ -category) and C ∗ Cat [ W − ]by [DT14, Thm. 1.4] are semi-additive, the functor Hg Morita then also preserves finitecoproducts. This implies that the first morphism in (14.5) is an equivalence, too. Henceassuming the claim we conclude that (14.5) and therefore (14.3) are equivalences.It remains to show the claim. So let ( (cid:96)
Morita ( D j )) j ∈ J be a finite family of objects of C ∗ Cat [ W − ]. Since every object in C ∗ Cat is Morita equivalent to an additively andidempotently complete object (apply e.g. the functor ( − ) (cid:93) from (14.1)) we can assumewithout loss of generality that D j is additively and idempotently complete for every j in J .Since such objects are fibrant in the model category structure of [DT14] we can concludethat the canonical map (cid:96) Morita (cid:0) (cid:89) j ∈ J D j (cid:1) → (cid:89) j ∈ J (cid:96) Morita ( D j ) (14.6)is an equivalence. Applying Hg Morita we get the equivalenceHg
Morita (cid:0) (cid:96)
Morita (cid:0) (cid:89) j ∈ J D j (cid:1)(cid:1) (cid:39) −→ Hg Morita (cid:0) (cid:89) j ∈ J (cid:96) Morita ( D j ) (cid:1) . (14.7)97ince D j is additive and hence admits a zero object for every j in J we can applyLemma 11.9 in order to conclude that the lower horizontal morphism in the commutativediagram Hg Morita ( (cid:96) Morita ( (cid:81) j D j )) ! (cid:47) (cid:47) (14.4) (cid:39) (cid:15) (cid:15) (cid:81) j Hg Morita ( (cid:96) Morita ( D j )) (14.4) (cid:39) (cid:15) (cid:15) Hg( (cid:81) j ∈ J D j ) Lem. 11 . (cid:47) (cid:47) (cid:81) j ∈ J Hg( D j )is an equivalence. Hence the arrow marked by ! is an equivalence. Composing this arrowwith the inverse of (14.7) provides the desired equivalenceHg Morita (cid:0) (cid:89) j ∈ J (cid:96) Morita ( D j ) (cid:1) (cid:39) → (cid:89) j Hg Morita ( (cid:96) Morita ( D j )) . This finishes the verification of the claim and therefore the proof of the lemma.
Remark 14.12.
If Hg : C ∗ Cat → M is a Morita invariant homological functor, then itin particular preserves the empty coproduct, i.e., the canonical map 0 M → Hg( ∅ ) is anequivalence.Let Hg : C ∗ Cat nu → M be a homological functor. Corollary 14.13. If Hg is Morita invariant and finitary (see Definition 11.6), then Hg preserves all small coproducts.Proof. Every small coproduct is a small filtered colimit over finite coproducts. Hence theclaim follows from the previous Lemma 14.11 and since Hg preserves small filtered colimitsby assumption.Recall Definition 12.3 of the functor K C ∗ Cat : C ∗ Cat → Sp . Theorem 14.14.
The functor K C ∗ Cat is Morita invariant.Proof.
We use the characterization of Morita invariant functors provided by Lemma 14.10.We already know that K C ∗ Cat sends unitary equivalences to equivalenence since it is ahomological functor.Let D → E be a functor satisfying the Conditions 2(b)i, 2(b)ii and 2(b)iii. We identify D with a full subcategory of E . Then we must show thatK C ∗ Cat ( D ) → K C ∗ Cat ( E )98s an equivalence. Note that the homomorphism of C ∗ -algebras A ( D ) → A ( E ) (see (4.1)for A ( − )) is defined since D → E is injective on objects. Since D → E is fully faithful,for every finite set F of objects F in D the composition A ( F ) → A ( D ) → A ( E ) is anembedding, see (13.4) for A ( F ).In view of Lemma 12.8 it suffices to show that the induced homomorphism φ : π ∗ K C ∗ ( A ( D )) → π ∗ K C ∗ ( A ( E ))between K -theory groups is an isomorphism. In view of Bott periodicity (Remark 12.2.4)is suffices to consider the cases ∗ = 0 and ∗ = 1.We will use the shorter notation K ∗ := π ∗ K C ∗ .surjectivity for ∗ = 0:Let p be in K ( A ( E )). We must show that p is in the image of φ : K ( A ( D )) → K ( A ( E )).Since E is additive, by Lemma 13.3.1 we can find an object E in E and a pair of projections P, ˜ P in End E ( E ) such that (cid:96) E, ∗ ([ P ] − [ ˜ P ]) = p , where (cid:96) E : End E ( E ) → A ( E ) is the canonical(in general non-unital) embedding (13.5), see also Remark 13.2 for notation.By the assumption on the functor D → E we can choose a family of objects ( D i ) i =1 ,...,m of D and an isometry u : E → (cid:76) mi =1 D i . For every i in { , . . . , m } we define the morphism u i := e ∗ i u : E → D i , where ( e i ) mi =1 is the family of structure maps for the sum (cid:76) mi =1 D i . Then the m × m -matrixwith entries in A ( E ) u (cid:48) := m (cid:88) i =1 u i [ i, Mat m ( A ( E )). We consider the finite subset F := { D , . . . , D m } ofobjects in D . The conjugation map u (cid:48) ( − ) u (cid:48) , ∗ : Mat m ( A ( E )) → Mat m ( A ( E )) has values inthe subalgebra Mat m ( A ( F )) of Mat m ( A ( E )), and u (cid:48) , ∗ u (cid:48) = h ( id E ), where h := (cid:15) A ( E ) ,n [1] ◦ (cid:96) E : End E ( E ) → Mat m ( A ( E ))and (cid:15) A ( E ) ,n [1] is as in (13.1). By construction we have p = (cid:15) A ( E ) ,n [1] − ∗ h ∗ ([ P ] − [ ˜ P ]) . We consider ˜ h := u (cid:48) hu (cid:48) , ∗ as a homomorphism from End E ( E ) to Mat m ( A ( F )) and let h (cid:48) be its composition with κ : Mat m ( A ( F )) → Mat m ( A ( D )) and Mat m ( A ( D )) → Mat m ( A ( E )).Then the chain of equalities p = (cid:15) A ( E ) ,n [1] − ∗ h ∗ ([ P ] − [ ˜ P ]) (13.2) = (cid:15) A ( E ) ,n [1] − ∗ h (cid:48)∗ ([ P ] − [ ˜ P ]) = φ ( (cid:15) A ( D ) ,n [1] − ∗ κ ∗ ˜ h ∗ ([ P ] − [ ˜ P ]))99hows that p is in the image of φ .injectivity for ∗ = 0:Let p be in K ( A ( D )) such that φ ( p ) = 0. As explained in Remark 13.2 and the beginningof the proof of Lemma 13.3 there exists a finite subset F of objects in D and projections P, ˜ P in Mat n ( A ( F )) such that p = κ ∗ ([ P ] − [ ˜ P ]), where κ : A ( F ) → A ( D ) is the inclusionand [ P ] , [ ˜ P ] are considered in K ( A ( F )).Using the inclusion A ( F ) → A ( E ) we can consider the projections P and ˜ P as elements in Mat n ( E ). Since φ ( p ) = 0, after increasing n if necessary there exists a partial isometry U in Mat n ( A ( E ) + ) such that U U ∗ = P and U ∗ U = ˜ P . These two equalities together implythat U belongs to the subalgebra Mat n ( A ( F )) of Mat n ( A ( E ) + ). Consequently, [ P ] = [ ˜ P ]and hence p = 0.surjectivity for ∗ = 1:Let u be in K ( A ( E )). Since E is additive, by Lemma 13.6.1 we can find an object E in E and a unitary U in End E ( E ) with (cid:96) E, ∗ [ U ] = u . Then as in the argument for surjectivityfor ∗ = 0 we have u = (cid:15) A ( E ) ,n [1] − ∗ h ∗ [ U ] = (cid:15) A ( E ) ,n [1] − ∗ h (cid:48)∗ [ U ] = φ ( (cid:15) A ( C ) ,n [1] − ∗ κ ∗ ˜ h ∗ ([ U ]))so that u is in the image of of φ .injectivity for ∗ = 1:Let u in K ( A ( D )) be such that φ ( u ) = 0. As in the proof of Lemma 13.6.1 there exists afinite set of objects F (cid:48) of D and n in N such that there is an unitary U in Mat n ( A ( F (cid:48) ))with [ U ] = u . We let [ U ] F (cid:48) ,n in K ( Mat n ( A ( F (cid:48) )) denote the corresponding class and κ F (cid:48) ,n : Mat n ( A ( F (cid:48) )) → Mat n ( A ( D )) be the inclusion. Then we have the equality u = [ U ] (13.13) = (cid:15) A ( D ) ,n [1] − ∗ κ F (cid:48) ,n, ∗ ([ U ] F (cid:48) ,n ) . (14.8)We can further find an object E in E and a homomorphism ψ : Mat n ( A ( F (cid:48) )) → End E ( E )such that φ ( u ) = (cid:96) E, ∗ ( ψ ∗ ([ U ] F (cid:48) ,n )).Since φ ( u ) = 0, by Lemma 13.6.2, after enlarging E if necessary, we can assume that wehave ψ ∗ [ U ] F (cid:48) ,n = 0.We let F (cid:48)(cid:48) be the union of the family F chosen above in order to represent E as a subobjectand the family F (cid:48) . Let now ˜ h : End E ( E ) → Mat m ( A ( F (cid:48)(cid:48) )) be as above. Then˜ h ∗ ψ ∗ [ U ] F (cid:48) ,n = 0 . (14.9)100y an inspection of the construction one observes that˜ h ◦ ψ : Mat n ( A ( F (cid:48) )) → Mat m ( A ( F (cid:48)(cid:48) ))is given by conjugation w ( − ) w ∗ by an element in w in Mat ( m, n, A ( F (cid:48)(cid:48) )) such that w ∗ w =1 A ( F (cid:48) ) ,n . This implies that ˜ h ∗ ψ ∗ : K ( Mat n ( A ( F (cid:48) ))) → K ( Mat n ( A ( F (cid:48)(cid:48) ))) is equal to themap induced by the inclusion ι : Mat n ( A ( F (cid:48) )) → Mat m ( A ( F (cid:48)(cid:48) )). Consequently u (14.8) = (cid:15) A ( D ) ,n [1] − ∗ κ F (cid:48) ,n, ∗ ([ U ] F (cid:48) ,n )= (cid:15) A ( D ) ,m [1] − ∗ κ F (cid:48)(cid:48) ,m, ∗ ι ∗ ([ U ] F (cid:48) ,n )= (cid:15) A ( D ) ,m [1] − ∗ κ F (cid:48)(cid:48) ,m, ∗ ˜ h ∗ ψ ∗ ([ U ] F (cid:48) ,n ) (14.9) = 0 . Corollary 14.15. K C ∗ Cat preserves all small coproducts.Proof.
By Theorem 14.14 the functor K C ∗ Cat is Morita invariant, and by Theorem 12.4 itis finitary. The claim now follows from Corollary 14.13.
Remark 14.16.
In [DT14, Rem. 10.12] the authors review various definitions of K -groupsfor C ∗ -categories appearing in the literature and compare them with their functor K D’A-T0 ( C ) := Hom Ho ( C ∗ Cat [ W − ] Idem ⊕ ) ( C (cid:93) , C (cid:93) )(see (14.1) for (cid:93) ) which is Morita invariant by definition. In particular in Point (iii) of thatremark they mention the version π K C ∗ Cat ( C ) considered in the present paper. It is notclear that these two K -functors are isomorphic.
15 Relative equivalences and idempotent completions
Note that the notions of a unitary equivalence or idempotent completion are only definedfor unital C ∗ -categories. In the following we extend these notions to the relative situationof an ideal in a unital C ∗ -category (Definition 11.1).We consider C in C ∗ Cat and an ideal K in C . Definition 15.1.
We define
Idem( K ) to be the wide subcategory of Idem( C ) of morphismsbelonging to K . Remark 15.2.
Unfolding the definition of the idempotent completion [DT14, Defn. 2.15],we get the following: 101
An object of Idem( C ), equivalently of Idem( K ), is a pair ( C, p ) of an object C of C and a projection p on C belonging to C (we explicitly do not require that p is in K ). • A morphism A : ( C, p ) → ( C (cid:48) , p (cid:48) ) in Idem( C ) is a morphism A : C → C (cid:48) such that p (cid:48) A = A = Ap . This morphism belongs to Idem( K ) if and only if A belongs to K .Note that Idem( K ) depends on the embedding of K into C .Let Hg : C ∗ Cat nu → M be a homological functor. Proposition 15.3. If Hg is Morita invariant, then the morphism Hg( K ) → Hg(Idem( K )) is an equivalence.Proof. We consider the exact sequence0 → K → C → Q → C ∗ Cat nu , where Q is defined to be the quotient. We then get an exact sequence0 → Idem( K ) → Idem( C ) → Idem C ( Q ) → C ( Q ) is again defined to be the quotient. We have a canonical fully faithfulfunctor Idem C ( Q ) → Idem( Q ) which sends ( C, p ) to ( C, [ p ]), and which is the obviousmap on morphisms. In order to see fully faithfulness note that if [ A ] : ( C, [ p ]) → ( C (cid:48) , [ p (cid:48) ]) isa morphism in Idem( Q ), then the relations [ p (cid:48) ][ A ] = [ A ] = [ A ][ p ] imply that [ A ] = [ p (cid:48) Ap ].Hence [ A ] can be lifted to a morphism p (cid:48) Ap : ( C, p ) → ( C (cid:48) , p (cid:48) ) in Idem( C ). Thus Idem C ( Q )is the full subcategory of Idem( Q ) consisting of objects ( C, [ p ]) such that [ p ] lifts to aprojection in C .We now consider the following commuting diagram:0 (cid:47) (cid:47) K (cid:47) (cid:47) !!! (cid:15) (cid:15) C (cid:47) (cid:47) ! (cid:15) (cid:15) Q (cid:47) (cid:47) !! (cid:15) (cid:15) ! (cid:115) (cid:115) (cid:15) (cid:15) (cid:47) (cid:47) Idem( K ) (cid:47) (cid:47) Idem( C ) (cid:47) (cid:47) Idem C ( Q ) (cid:47) (cid:47) ! (cid:15) (cid:15) Q )The arrows marked by ! are Morita equivalences. They induce equivalences after applyingHg. This implies that also !! induces an equivalence after applying Hg. Applying Hg tothe horizontal sequences yields fibre sequences in M . By the Five Lemma we can thenconclude that also the arrow marked by !!! induces an equivalence after applying Hg.We now consider two objects K , K (cid:48) in C ∗ Cat nu and a functor f : K → K (cid:48) .102 efinition 15.4. We call f a relative equivalence if there exists a pull-back diagram K (cid:47) (cid:47) f (cid:15) (cid:15) C ˆ f (cid:15) (cid:15) K (cid:48) (cid:47) (cid:47) C (cid:48) such that the horizontal morphisms are inclusions of ideals and ˆ f an equivalence in C ∗ Cat . Let Hg : C ∗ Cat nu → M be a homological functor. Proposition 15.5. If f : K → K (cid:48) is a relative equivalence, then Hg( f ) : Hg( K ) → Hg( K (cid:48) ) is an equivalence.Proof. We have the following morphism0 (cid:47) (cid:47) K (cid:47) (cid:47) f (cid:15) (cid:15) C (cid:47) (cid:47) ˆ f (cid:15) (cid:15) Q (cid:47) (cid:47) ¯ f (cid:15) (cid:15) (cid:47) (cid:47) K (cid:48) (cid:47) (cid:47) C (cid:48) (cid:47) (cid:47) Q (cid:48) (cid:47) (cid:47) C ∗ Cat nu , where Q and Q (cid:48) are defined as the quotients. Since ˆ f isfully faithful and the left square is a pull-back, we see that ¯ f is also fully faithful. Since ˆ f is essentially surjective, so is ¯ f . Hence ¯ f is an equivalence, too.We now use that Hg sends the horizontal sequences to fibre sequences and the two rightvertical morphisms to equivalences. By the Five Lemma we conclude that it also sendsthe left vertical morphism to an equivalence.Let f, g : C → D two functors in C ∗ Cat , and let Hg : C ∗ Cat → M be a functor. Lemma 15.6. If Hg sends unitary equivalences to equivalences and f and g are unitarilyisomorphic, then we have an equivalence Hg( f ) (cid:39) Hg( g ) .Proof. We define a category E in C ∗ Cat as follows.1. objects: Ob( E ) := Ob( C ) (cid:116) Ob( C ). For C in C we let C and C denote the twocopies of C in E .2. morphisms: For C, C (cid:48) in C and i, j in { , } we set Hom E ( C i , C (cid:48) j ) := Hom C ( C, C (cid:48) ).3. composition and involution are defined in the obvious way.103e have two inclusions ι , ι : C → E sending C to C and C , respectively. We furtherhave a projection p : E → C defined in the obvious way.Note that p ◦ ι = p ◦ ι = id C . Furthermore, we have unitary isomorphisms v i : ι i ◦ p → id .For example, v is given by v ,C = id C and v ,C = id C in Hom E ( C , C ).We conclude that p is a unitary equivalence and ι , ι are both unitary equivalences whichare inverse to p . In particular we have an equivalence Hg( ι ) (cid:39) Hg( ι ).Let u : f → g be the unitary isomorphism. We then define a functor h : E → D as follows:1. objects: For C in C we set h ( C ) := f ( C ) and h ( C ) := g ( C ).2. morphisms: We distinguish the following four cases.a) c : C → C (cid:48) is sent to h ( c ) := f ( c ).b) c : C → C (cid:48) is sent to h ( c ) := u C (cid:48) f ( c ).c) c : C → C (cid:48) is sent to h ( c ) := u ∗ C (cid:48) g ( c ).d) c : C → C (cid:48) is sent to h ( c ) := g ( c ).One checks that this defines a morphism in C ∗ Cat . We note that h ◦ ι = f and h ◦ ι = g .We now conclude Hg( f ) (cid:39) Hg( h ) ◦ Hg( ι ) (cid:39) Hg( h ) ◦ Hg( ι ) (cid:39) Hg( g ) . We again consider the relative situation. Let f, g : K → L be two functors in C ∗ Cat nu . Definition 15.7.
We say that f and g are relatively unitarily isomorphic if there existsan ideal inclusion ω : L → D for some D in C ∗ Cat such that the compositions ω ◦ f and ω ◦ g are unitarily isomorphic. Let Hg : C ∗ Cat → M be a functor and f, g : K → L be two functors in C ∗ Cat nu . Lemma 15.8. If Hg is a homological functor and f and g are relatively unitarily isomor-phic, then Hg( f ) (cid:39) Hg( g ) .Proof. We let C := K + be the unitalization of K and ˆ f , ˆ g : C → D be the canonicalunital extensions of ω ◦ f and ω ◦ g . Then ˆ f and ˆ g are unitarily isomorphic. We consider104he morphisms of exact sequences0 (cid:47) (cid:47) K (cid:47) (cid:47) f,g (cid:15) (cid:15) C (cid:47) (cid:47) ˆ f, ˆ g (cid:15) (cid:15) Q (cid:47) (cid:47) ¯ f, ¯ g (cid:15) (cid:15) (cid:47) (cid:47) L ω (cid:47) (cid:47) D (cid:47) (cid:47) R (cid:47) (cid:47) f , resp. ˆ g . Here Q and R are defined as quotients. The unitary isomorphismbetween ˆ f and ˆ g induces a unitary isomorphism between ¯ f and ¯ g . If we use the constructionof the equivalences Hg( ˆ f ) (cid:39) Hg(ˆ g ) and Hg( ¯ f ) (cid:39) Hg(¯ g ) as in the proof of Lemma 15.6, weobtain an equivalence between the following commutative squares:Hg( C ) (cid:47) (cid:47) Hg( ˆ f ) (cid:15) (cid:15) Hg( Q ) Hg( ¯ f ) (cid:15) (cid:15) Hg( D ) (cid:47) (cid:47) Hg( R ) Hg( C ) (cid:47) (cid:47) Hg(ˆ g ) (cid:15) (cid:15) Hg( Q ) Hg(¯ g ) (cid:15) (cid:15) Hg( D ) (cid:47) (cid:47) Hg( R )Note that this includes also a higher relation between fillers. We now functorially extendthese squares horizontally to morphisms between fibre sequences. Using the assumptionthat Hg is a homological functor (namely that it sends exact sequences to fibre sequencesand unitary equivalences to equivalences) we see that the equivalence between the twocommutative squares then induces an equivalence Hg( f ) (cid:39) Hg( g ).In the next proposition we weaken the assumption in Lemma 15.7 from relatively unitarilyisomorphic to relatively Murray–von Neumann equivalent. We start with defining thisnotion for a pair of functors f, g : K → L in C ∗ Cat nu . Definition 15.9.
We say that f and g are relatively Murray–von Neumann equivalent(MvN equivalent) if there exists an ideal inclusion ω : L → D for some D in C ∗ Cat anda natural transformation u : ω ◦ f → ω ◦ g given by u = ( u C ) C ∈ K , where u C is a partialisometry in D for every object C of K such that u ∗ C (cid:48) u C (cid:48) f ( k ) = f ( k ) and g ( k ) u C u ∗ C = g ( k ) for all morphisms k : C → C (cid:48) in K . We consider two functors f, g : K → L in C ∗ Cat nu and a functor Hg : C ∗ Cat → M . Proposition 15.10. If f and g are relatively MvN equivalent and Hg is a Morita invarianthomological functor, then Hg( f ) (cid:39) Hg( g ) .Proof. Let u = ( u C ) C ∈ Ob( K ) : f → g be the natural transformation implementing the MvNequivalence between the functors f and g considered as functors with values in D (we drop ω from the notation for better readability). For every object C of K we have projections p C := u ∗ C u C on f ( C ) and q C := u C u ∗ C on g ( C ) belonging to D .105f k : C → C (cid:48) is a morphism in K , then we have p C (cid:48) f ( k ) = f ( k ) by assumption. Note also f ( k ) p C = ( f ( k ) p C ) ∗∗ = ( p C f ( k ∗ )) ∗ = f ( k ∗ ) ∗ = f ( k ) . Recall that Idem( L ) is defined relatively to the ideal inclusion of L into D . The functor f canonically induces a functor ˜ f : K → Idem( L ) given as follows:1. objects: The functor ˜ f sends the object C in K to the object ( f ( C ) , p C ) of Idem( L ).2. morphisms: The functor ˜ f sends a morphism k : C → C (cid:48) in K to the morphism f ( k ) : ( f ( C ) , p C ) → ( f ( C (cid:48) ) , p C (cid:48) ).The observations above show that this functor is well-defined. We have a similarly definedfunctor ˜ g : K → Idem( L ).We let Emb : Idem( L ) → Idem( L ) be the endofunctor given as follows:1. objects: It sends the object ( D, p ) in Idem( L ) to the object D = ( D, id D ) of Idem( L ).2. morphisms: The functor Emb sends a morphism φ : ( D, p ) → ( D (cid:48) , p (cid:48) ) in Idem( L ) tothe morphism φ : D → D (cid:48) in Idem( L ).We have the following commuting diagram K f (cid:47) (cid:47) ˜ f (cid:15) (cid:15) L c (cid:15) (cid:15) Idem( L ) Emb (cid:47) (cid:47)
Idem( L ) (15.1)where c is the canonical inclusion. We have a similar diagram for g .We now note that u defines a relative unitary isomorphism ˜ u : ˜ f → ˜ g (for the inclusion L → Idem( D )): Indeed, we have ˜ u = (˜ u C ) C ∈ Ob( K ) , where˜ u C = q C u C p C : ( f ( C ) , p C ) → ( g ( C ) , q C )is a unitary isomorphism in Idem( D ). By Lemma 15.8 we conclude Hg( ˜ f ) (cid:39) Hg(˜ g ). Thisimplies Hg(Emb ◦ ˜ f ) (cid:39) Hg(Emb ◦ ˜ g ). Applying Hg to the commuting square (15.1) thisequivalence implies the equivalence Hg( c ◦ f ) (cid:39) Hg( c ◦ g ). Since we assume that Hg isMorita invariant we know by Proposition 15.3 that Hg( c ) is an equivalence. Hence weconclude Hg( f ) (cid:39) Hg( g ). 106 In this section we introduce the notion of a weak Morita equivalence in C ∗ Cat nu andshow that a weak Morita equivalence induces an equivalence in K -theory. In contrast tothe algebraic notion of Morita equivalences as introduced in Section 14 the weak Moritaequivalences are of analytic nature. They involve the possibility of norm-approximatingmorphisms in a larger category by morphisms in a smaller one. The typical example of aweak Morita equivalence is the left upper corner inclusion of C into the compact operatorson a Hilbert space which is considered as a functor between single-object C ∗ -categories.Let D be in C ∗ Cat nu . Recall that D u denotes the subcategory of unital objects of D . Let S be a subset of the set of objects of D u . Definition 16.1. S is weakly generating if for every D in D , any finite family ( f i ) i ∈ I ofmorphisms f i : D i → D in D , and any ε in (0 , ∞ ) there exists an isometry u : C → D in D such that (cid:107) f i − uu ∗ f i (cid:107) ≤ ε for all i in I and C is unitarily isomorphic to a finiteorthogonal sum of objects in S . Let C , D be in C ∗ Cat nu , and let φ : C → D be a functor. Definition 16.2.
The functor φ is a weak Morita equivalence if it has the followingproperties:1. C is unital.2. φ is fully faithful.3. φ (Ob( C )) is weakly generating. Remark 16.3.
The notion of a weak Morita equivalence should not be confused with thenotion of a Morita equivalence. In general, a Morita equivalence need not be a weak Moritaequivalence or vice versa. Our motivation to use the term
Morita also in this situation isthat a weak Morita equivalence φ : C → D gives rise to a Morita ( A ( C ) , A ( D ))-bi-modulewhich is at the heart of the proof of Theorem 16.4. Theorem 16.4. If φ is a weak Morita equivalence, then K C ∗ Cat ( φ ) : K C ∗ Cat ( C ) → K C ∗ Cat ( D ) is an equivalence. The theorem has the following consequence which has already been observed in [Joa03]and [Mit01]. Assume that φ : C → D is a morphism in C ∗ Cat .107 orollary 16.5. If φ : C → D is a unitary equivalence, then K C ∗ Cat ( φ ) is an equivalence.Proof. We show that φ is a weak Morita equivalence. By assumption, C is unital. Since φ is an equivalence it is fully faithful. It remains to show that φ (Ob( C )) is weakly generating.In this case we have a much stronger property: Let D be an object of D . Since φ isessentially surjective, there exists C in C and a unitary u : φ ( C ) → D (cid:48) . Then for any f : D (cid:48) → D we have f = uu ∗ f . Remark 16.6.
The specialization of the proof of the Theorem 16.4 to the special caseconsidered in Corollary 16.5 is essentially equivalent to the proof given in [Joa03].The idea of the proof of Theorem 16.4 is to reduce the assertion to the Morita invarianceof the K -theory of C ∗ -algebras. We first recall some of the basic facts.Let A and B be in C ∗ Alg nu . Recall that a Hilbert B -module ( H, (cid:104)− , −(cid:105) B ) is called full if (cid:104) H, H (cid:105) B is dense in B . Definition 16.7.
A Morita ( A, B ) -bimodule is a triple ( H, A (cid:104)− , −(cid:105) , (cid:104)− , −(cid:105) B ) , where H isan ( A, B ) -bimodule, A (cid:104)− , −(cid:105) is an A -valued scalar product on H and (cid:104)− , −(cid:105) B is a B -valuedscalar product on H such that1. ( H, (cid:104)− , −(cid:105) B ) is a full Hilbert B -module.2. ( H, A (cid:104)− , −(cid:105) ) is a full Hilbert A -module.3. For all h, h (cid:48) , h (cid:48)(cid:48) in H we have the relation A (cid:104) h, h (cid:48) (cid:105) h (cid:48)(cid:48) = h (cid:104) h (cid:48) , h (cid:48)(cid:48) (cid:105) B . (16.1) Remark 16.8.
The datum of a Morita (
A, B )-bimodule is equivalent to the datum of atriple ( H, (cid:104)− , −(cid:105) B , φ ) of a Hilbert B -module ( H, (cid:104)− , −(cid:105) B ) together with a homomorphism φ : A → B ( H ) such that1. ( H, (cid:104)− , −(cid:105) B ) is full.2. φ is an isomorphism from A to K ( H ).In this case one can reconstruct the A -valued scalar product by A (cid:104) h, h (cid:48) (cid:105) := φ − (Θ h,h (cid:48) ),where Θ h,h (cid:48) is the one-dimensional operator on H acting as h (cid:48)(cid:48) (cid:55)→ Θ h,h (cid:48) ( h (cid:48)(cid:48) ) = h (cid:104) h (cid:48) , h (cid:48)(cid:48) (cid:105) B .In the other direction, assuming the data in Definition 16.7, the relationΘ h,h (cid:48) ( h (cid:48)(cid:48) ) = h (cid:104) h (cid:48) , h (cid:48)(cid:48) (cid:105) B = A (cid:104) h, h (cid:48) (cid:105) h (cid:48)(cid:48) h,h (cid:48) is given by the multiplication by an element of A . This extends to anisomorphism φ between A and K ( H ). Definition 16.9.
The datum of a Morita ( A, B ) -bimodule is called a strong Morita–Rieffelequivalence between A and B . Remark 16.10.
A strong Morita–Rieffel equivalence between A and B induces an equiv-alence Hilb ( A ) fg , proj (cid:51) M (cid:55)→ M ⊗ A H ∈ Hilb ( B ) fg , proj (16.2)of the topologically enriched categories of finitely generated, projective modules over A and B . It is possible to construct the topological K -theory spectrum of C ∗ -algebras fromthis category in a functorial way. Using such a construction in the background, a strongMorita–Rieffel equivalence between A and B gives rise to an equivalence between K -theoryspectra K C ∗ ( A ) → K C ∗ ( B ). We will not go into this direction since in the present paper weuse the K -theory of C ∗ -algebras in an axiomatic way and therefore only have functorialiyfor homomorphisms between C ∗ -algebras.Let f : A → B be a morphism in C ∗ Alg nu . Definition 16.11.
We say that f induces a strong Morita–Rieffel equivalence if thefollowing conditions are satisfied:1. H := f ( A ) B with the B -valued scalar product given by ( b, b (cid:48) ) (cid:55)→ b ∗ b (cid:48) is a full rightHilbert B -module.2. f : A → End B ( H ) identifies A with K ( H ) . In view of Remark 16.8 the homomorphism f gives rise to a strong Morita–Rieffel equiva-lence between A and B . Lemma 16.12. If f : A → B induces a strong Morita–Rieffel equivalence, then the inducedmorphism K C ∗ ( f ) : K C ∗ ( A ) → K C ∗ ( B ) is an equivalence.Proof. Using Bott periodicity it suffices to check that K C ∗ ∗ ( f ) : K C ∗ ∗ ( A ) → K C ∗ ∗ ( B ) is anisomorphism for ∗ = 0 ,
1. The point is now that the well-known isomorphism betweenK C ∗ ∗ ( A ) and K C ∗ ∗ ( B ) induced by the Morita ( A, B )-bimodule given in Definition 16.11 isprecisely the homomorphism K C ∗ ∗ ( f ). 109 roof of Theorem 16.4. We first assume that φ : C → D is injective on objects. Then wehave a commuting diagram K C ∗ Cat ( C ) K C ∗ Cat ( φ ) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) K C ∗ Cat ( D ) (cid:39) (cid:15) (cid:15) K C ∗ ( A ( C )) K C ∗ ( A ( φ )) (cid:47) (cid:47) K C ∗ ( A ( D ))where the vertical equivalences are induced by the natural transformation A f → A , seeLemma 12.8. The assumption on φ is needed since A is only functorial for morphismswhich are injective on objects. It suffices to show that K C ∗ ( A ( φ )) is an equivalence.We claim that K C ∗ ( A ( φ )) induces a strong Morita–Rieffel equivalence. Recall that A ( D )is a closure of the matrix algebra A alg ( D ) = (cid:77) D,D (cid:48) ∈ D Hom D ( D (cid:48) , D ) . It is an ( A alg ( C ) , A alg ( D ))-bimodule. Using that φ is injective on objects, we can considerthe ( A alg ( C ) , A alg ( D ))-bimodule H alg := (cid:77) C ∈ C ,D ∈ D Hom D ( D, φ ( C )) . as a sub-bimodule of A alg ( D ). Its elements will be written as families ( h C,D ) C ∈ C ,D ∈ D with finitely many non-zero members. A similar notation will be used for the elements of A alg ( C ) and A alg ( D ). The action of A alg ( C ) is given by( ah ) CD := (cid:88) C (cid:48) ∈ C φ ( a CC (cid:48) ) h C (cid:48) D for all a in A alg ( C ) and h in H alg . Similarly, the action of A alg ( D ) is given by( hb ) CD := (cid:88) D (cid:48) ∈ D h CD (cid:48) b D (cid:48) D for all h in H alg and b in A alg ( D ). In this notation the A ( D )-valued scalar product is givenby ( (cid:104) h, h (cid:48) (cid:105) A ( D ) ) D (cid:48) D := (cid:88) C ∈ D h ∗ CD (cid:48) h (cid:48) CD for all h, h (cid:48) in H alg . Furthermore, we define an A alg ( C )-valued scalar product by( A ( C ) (cid:104) h, h (cid:48) (cid:105) ) C (cid:48) C := φ − (cid:0) (cid:88) D ∈ D h C (cid:48) D h (cid:48) , ∗ CD (cid:1) for all h, h (cid:48) in H alg . Here we use that φ is fully faithful.110ne checks the relation A ( C ) (cid:104) h, h (cid:48) (cid:105) h (cid:48)(cid:48) = h (cid:104) h (cid:48) , h (cid:48)(cid:48) (cid:105) A ( D ) (16.3)for all h, h (cid:48) , h (cid:48)(cid:48) in H alg .We let H be the closure of H alg with respect to the norm induced by the A ( D )-valued scalarproduct, or equivalently, the closure in A ( D ). Then H is a right Hilbert A ( D )-module. Inthe notation of Definition 16.11 this is A ( φ ) A ( D ).We claim that the scalar product A ( C ) (cid:104)− , −(cid:105) on H alg extends by continuity to an A ( C )-valued scalar product on H . The relation (16.3) implies (cid:107) A ( C ) (cid:104) h, h (cid:48) (cid:105) h (cid:48)(cid:48) (cid:107) = (cid:107) h (cid:104) h (cid:48) , h (cid:48)(cid:48) (cid:105) A ( D ) (cid:107) ≤ (cid:107) h (cid:107)(cid:107) h (cid:48) (cid:107)(cid:107) h (cid:48)(cid:48) (cid:107) . All these norms are defined using the A ( D )-valued scalar product and the norm in A ( D ).Hence we can estimate the operator norm of A ( C ) (cid:104) h, h (cid:48) (cid:105) on H by (cid:107) A ( C ) (cid:104) h, h (cid:48) (cid:105)(cid:107) ≤ (cid:107) h (cid:107)(cid:107) h (cid:48) (cid:107) . (16.4)For every C (cid:48) in C the module H contains the closed ( A alg ( C ) , End D ( φ ( C (cid:48) )))-submodule gen-erated by (cid:76) C ∈ C Hom D ( φ ( C (cid:48) ) , φ ( C )). This module is isomorphic to the ( A alg ( C ) , End C ( C (cid:48) ))-module generated by (cid:76) C ∈ C Hom C ( C (cid:48) , C ) (again since φ is fully faithful). It is known by[Joa03] that the maximal norm on A ( C ) is induced by the representions of A alg ( C ) on thefamily of these modules for running C (cid:48) in Ob( C ). It follows that the operator norm on H induces the norm on A alg ( C ). The estimate (16.4) now implies that A ( C ) (cid:104)− , −(cid:105) extends bycontinuity to an A ( C )-valued scalar product on H . Furthermore, the action of A alg ( C ) on H alg extends to an action of A ( C ) on H such that H is a pre-Hilbert A ( C )-left module.We now show that A ( C ) = A ( C ) (cid:104) H, H (cid:105) . Let [ f CC (cid:48) ] be a one-entry matrix in A ( C ). We consider the one-entry matrices [ h Cφ ( C ) ] with h Cφ ( C ) := φ ( id C ) (note that we assume that C is unital) and [ h (cid:48) C (cid:48) φ ( C ) ] := φ ( f ∗ CC (cid:48) ) in H .Then A ( C ) (cid:104) [ h Cφ ( C ) ] , [ h (cid:48) C (cid:48) φ ( C ) ] (cid:105) = [ f CC (cid:48) ]. We now use that A ( C ) is generated by one-entrymatrices to show the claim.We now show that A ( D ) = (cid:104) H, H (cid:105) A ( D ) . Let f : D (cid:48) → D be a morphism in D such that there is an object C in C and an isometry u : φ ( C ) → D such that f = uu ∗ f . We will call such an element special. Let [ f DD (cid:48) ] be aone-entry matrix in A alg ( D ) with f DD (cid:48) = f . Then we consider the one-element matrices[ h CD ] in H with h CD := u ∗ and [ h (cid:48) CD (cid:48) ] in H with h (cid:48) CD (cid:48) := u ∗ f . Then (cid:104) [ h CD ] , [ h (cid:48) CD (cid:48) ] (cid:105) A ( D ) = [ f DD (cid:48) ] . We claim that one-element matrices with special entries generate A ( D ). It suffices to showthat special elements generate a dense subspace of Hom D ( D (cid:48) , D ) for all objects D, D (cid:48) in D .111e consider f : D (cid:48) → D and ε in (0 , ∞ ). Since φ (Ob( C )) is weakly generating there exists afinite family ( C i ) i ∈ I of objects in C , the orthogonal sum ( (cid:76) i ∈ I φ ( C i ) , ( e i ) i ∈ I ), and a partialisometry u : (cid:76) i ∈ I φ ( C i ) → D such that (cid:107) f − uu ∗ f (cid:107) ≤ ε . Then uu ∗ f = (cid:80) i ∈ I ue i e ∗ i u ∗ f .The summands ue i e ∗ i u ∗ f are special. Hence uu ∗ f is a finite sum of special elements.We now show that the pre-Hilbert A ( C )-module H is actually a Hilbert A ( C )-module.We let (cid:107) − (cid:107) (cid:48) denote the norm on H induced by the A ( C )-valued scalar product. We willshow that (cid:107) − (cid:107) is equivalent to (cid:107) − (cid:107) (cid:48) , where (cid:107) − (cid:107) is the norm on H induced by the A ( D )-valued scalar product. We then use that H is complete with respect to (cid:107) − (cid:107) byconstruction.From (16.4) we get the estimate (cid:107) − (cid:107) (cid:48) ≤ (cid:107) − (cid:107) . By (16.3) we get (cid:107) h (cid:104) h (cid:48) , h (cid:48) (cid:105) A ( D ) (cid:107) = (cid:107) A ( C ) (cid:104) h, h (cid:48) (cid:105) h (cid:48) (cid:107) ≤ (cid:107) h (cid:48) (cid:107)(cid:107) A ( C ) (cid:104) h, h (cid:48) (cid:105)(cid:107) ≤ (cid:107) h (cid:48) (cid:107)(cid:107) h (cid:107) (cid:48) (cid:107) h (cid:48) (cid:107) (cid:48) . Taking the supremum over all h in H with (cid:107) h (cid:107) ≤ (cid:107)(cid:104) h (cid:48) , h (cid:48) (cid:105) A ( D ) (cid:107) (cid:48)(cid:48) ≤ (cid:107) h (cid:48) (cid:107)(cid:107) h (cid:48) (cid:107) (cid:48) , (16.5)where (cid:107) − (cid:107) (cid:48)(cid:48) is the norm on A ( D ) induced from the operator norm on H . We claim that (cid:107) − (cid:107) (cid:48)(cid:48) is equal to the norm of A ( D ). The claim together with (16.5) then implies that (cid:107) h (cid:48) (cid:107) ≤ (cid:107) h (cid:48) (cid:107) (cid:48) for all h in H and hence (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) (cid:48) .We now show the claim that (cid:107) − (cid:107) (cid:48)(cid:48) is equal to the norm of A ( D ). Let b be in A alg ( D ) suchthat (cid:107) b (cid:107) = 1. We have to show that (cid:107) b (cid:107) (cid:48)(cid:48) = 1. For every D (cid:48) in D we let M D (cid:48) be the rightHilbert A ( D )-module generated by (cid:76) D ∈ D Hom D ( D, D (cid:48) ). It is a direct summand of A ( D ).We choose ε in (0 , ∞ ). By [Joa03] the family of modules ( M D (cid:48) ) D (cid:48) ∈ D induces the norm on A ( D ). Hence there exists D (cid:48) in D (cid:48) and m in M D (cid:48) , such that (cid:107) m (cid:107) ≤ (cid:107) mb (cid:107) ≥ − ε/ R of non-zero members of the family m = ( m D (cid:48) D ) D ∈ D is finite.We furthermore have (cid:107) m D (cid:48) D (cid:107) ≤ D in D . There exists a finite family of objects( C i ) i ∈ I in C , an object ( E, ( e i ) i ∈ I ), e i : φ ( C i ) → E , representing the orthogonal sum of thefamily ( φ ( C i )) i ∈ I , and an isometry u : E → D (cid:48) such that (cid:107) m D (cid:48) D − uu ∗ m D (cid:48) D (cid:107) ≤ ε R +1) forall D in D . Then (cid:107) m − uu ∗ m (cid:107) ≤ ε/
2. We consider the right A ( D )-Hilbert module M E .We note that u induces an isometry M E → M D (cid:48) . We set m (cid:48) := u ∗ m in M E . Then we have (cid:107) m (cid:48) b (cid:107) = (cid:107) um (cid:48) b (cid:107) = (cid:107) uu ∗ mb (cid:107) ≥ (cid:107) mb (cid:107) − (cid:107) ( m − uu ∗ m ) b (cid:107) ≥ − ε . For every i in I we have an isometric inclusion of right Hilbert A ( D )-modules f i : M φ ( C i ) → H sending ( m D ) D ∈ D to ( e i m D ) D ∈ D . Hence we get an isometric inclusion f := ⊕ i ∈ I f i : M E → (cid:77) i ∈ I H .
The diagonal representation of A ( D ) on (cid:76) i ∈ I H induces the same norm as the representa-tion on H . We then have (cid:107) f ( m (cid:48) ) b (cid:107)≥ − ε . Since (cid:107) hm (cid:48) (cid:107) ≤ (cid:107) b (cid:107) (cid:48)(cid:48) ≥ − ε .Since ε was arbitrary we conclude that (cid:107) b (cid:107) (cid:48)(cid:48) = 1.112his finishes the verification that ( H, A ( C ) (cid:104)− , −(cid:105) , (cid:104)− , −(cid:105) A ( D ) ) is an ( A ( C ) , A ( D ))-Moritabimodule, and that K C ∗ ( A ( φ )) induces a strong Morita–Rieffel equivalence. By Lemma16.12 we can conclude that K C ∗ ( A ( φ )) : K C ∗ ( A ( C )) → K C ∗ ( A ( D )) is an equivalence. Thisimplies the assertion of Theorem 16.4 and therefore also of Corollary 16.5 for functors φ which are injective on objects.We finally drop the assumption that φ is injective on objects. Let φ : C → D be a weakMorita equivalence. Then we form E in C ∗ Cat nu as follows:1. objects: The set of objects of E is given by Ob( C ) (cid:116) Ob( D ).2. morphisms: Hom E ( E, E (cid:48) ) :=
Hom C ( E, E (cid:48) ) for
E, E (cid:48) ∈ C , Hom D ( φ ( E ) , E (cid:48) ) for E ∈ C , E (cid:48) ∈ D , Hom D ( E, φ ( E (cid:48) )) for E ∈ D , E (cid:48) ∈ C , Hom D ( E, E (cid:48) ) for
E, E (cid:48) ∈ D .
3. composition, involution, G -action: these structures are defined in the canonical way.We have inclusions i : C → E , j : D → E and a projection p : E → D such that p ◦ j = id D and p ◦ i = φ . Moreover, there is anobvious unitary isomorphism j ◦ p ∼ = id E . We conclude that p is an equivalence. Moreover, i is again a weak Morita equivalence which is in addition injective on objects. By the specialcase already shown, K C ∗ Cat ( i ) is an equivalence. Hence K C ∗ Cat ( φ ) (cid:39) K C ∗ Cat ( p ) ◦ K C ∗ Cat ( i )is an equivalence. Example 16.13.
Let A be in C ∗ Alg and consider the wide subcategory K ( A ) of compactmorphisms in Hilb ( A ), cf. Example 2.7. Recall the Definition 2.10 of the subcategoryof unital objects K ( A ) u in K ( A ). An A -Hilbert module is in K ( A ) u if and only if it isalgebraically finitely generated, and all such modules are projective [WO93, Ex. 15.O andCor. 15.4.8].We let Hilb ( A ) std be the full subcategory of Hilb ( A ) of objects which are isomor-phic to (not necessarily finite) orthogonal sums of families of objects of K ( A ) u andset K ( A ) std := K ( A ) ∩ Hilb ( A ) std . Note that K ( A ) u std = K ( A ) u . We further have thefollowing commutative diagram of inclusion functors A weak Morita (cid:47) (cid:47) Morita (cid:34) (cid:34) K ( A ) std . K ( A ) u weak Morita (cid:57) (cid:57) (16.6)113pplying K -theory and using the Theorems 14.14 and 16.4 we obtain equivalencesK C ∗ ( A ) (cid:39) (cid:47) (cid:47) (cid:39) (cid:39) (cid:39) K C ∗ Cat ( K ( A ) std ) . K C ∗ Cat ( K ( A ) u ) (cid:39) (cid:53) (cid:53) (16.7)These equivalences will be used in companion paper [BEL].
17 Functors on the orbit category
For a group G we consider the orbit category G Orb of transitive G -sets and equivariantmaps. It plays a fundamental role in G -equivariant homotopy theory: by Elmendorf’stheorem [Elm83] (and subsequent work thereon) the category PSh ( G Orb ) models theequivariant homotopy theory of G -topological spaces. For a cocomplete ∞ -category M the ∞ -category of M -valued equivariant homology theories (possibly unstable) is equivalentto the ∞ -category of functors from G Orb to M , see Section 17.1. Such functors are themain ingredients of assembly maps, see e.g. [BEKW20c, Sec. 1] for more information.The present section deals with the construction of such functors from C ∗ -categories with G -action.Our first construction uses the homotopy theory of unital C ∗ -categories modeled by theDwyer–Kan localization (cid:96) : C ∗ Cat → C ∗ Cat ∞ of C ∗ Cat at the unitary equivalences, see Section 8 and [Bun19]. We can consider the set G with the left action as an object of G Orb . The right-action of G on itself induces anisomorphism of monoids G ∼ = End G Orb ( G ). We therefore have an embedding of categories j G : BG → G Orb (17.1)which sends the unique object ∗ BG of BG to the G -set ˜ G .Given a unital C ∗ -category with G -action ˜ D in Fun ( BG, C ∗ Cat ) we can define a functor j G ! ( (cid:96) BG ( ˜ D )) : G Orb → C ∗ Cat ∞ , (17.2)where j G ! is the left Kan extension functor along j G and the functor (cid:96) BG : Fun ( BG, C ∗ Cat ) → Fun ( BG, C ∗ Cat ∞ )is given by post-composition with (cid:96) . At this point the order of applying the localizationand the Kan extension is crucial in order to get the desired values. If H is a subgroup of G , then one can calculate the value of the functor (17.2) at the object G/H in G Orb .114 emma 17.1.
We have an equivalence j G ! ( (cid:96) BG ( ˜ D ))( G/H ) (cid:39) (cid:96) ( ˜ D (cid:111) H ) . Proof.
We use the pointwise formula for the left Kan extension which gives j G ! ( (cid:96) BG ( ˜ D ))( G/H ) (cid:39) colim BG /G/H (cid:96) BG ( ˜ D ) . We consider the functor BH → BG /G/H which sends the unique object ∗ BH of BH to the projection map G → G/H considered as a object of the slice category BG /G/H ,and the morphism h in H = End BH ( ∗ BH ) to the endomorphism of G → G/H given byright-multiplication with h − on G . This functor is an equivalence of categories. We cantherefore replace the slice category in the index of the colimit by BH . We further observethat the restriction of the functor (cid:96) BG ( ˜ D ) along BH → BG /G/H is given by (cid:96) BH (Res GH ( ˜ D )),where Res GH : Fun ( BG, C ∗ Cat ) → Fun ( BH, C ∗ Cat ) is the restriction of the group action.In [Bun, Thm. 7.8] we have seen that colim BH (cid:96) BH (Res GH ( ˜ D )) (cid:39) (cid:96) ( ˜ D (cid:111) H ) , where − (cid:111) H denotes the maximal crossed product. Combining the displayed equivalenceswe get the equivalence asserted in the lemma. Remark 17.2.
For most constructions in this first part we could start with an object ˜ D ∞ in Fun ( BG, C ∗ Cat ∞ ) in place if (cid:96) BG ( ˜ D ). The reason to prefer to start with the choice of˜ D in Fun ( BG, C ∗ Cat ) is precisely that in this case by Lemma 17.1 we have a preferredrepresentative of j G ! ( (cid:96) BG ( ˜ D )) given by the maximal crossed product. See Remark 17.4 fora continuation.Let Hg : C ∗ Cat → M be a functor which sends unitary equivalences to equivalences. Ourmain example is the restriction of K C ∗ Cat to unital C ∗ -categories. By the universal propertyof the Dwyer–Kan localization it has an essentially unique factorization C ∗ Cat Hg (cid:47) (cid:47) (cid:96) (cid:38) (cid:38) M .C ∗ Cat ∞ Hg ∞ (cid:57) (cid:57) (17.3) Definition 17.3.
We define the functor Hg G ˜ D , max := Hg ∞ ( j G ! ( (cid:96) BG ( ˜ D ))) : G Orb → M . (17.4)By Lemma 17.1 its value on the orbit G/H is given byHg G ˜ D , max ( G/H ) (cid:39) Hg( ˜ D (cid:111) H ) . (17.5)115 emark 17.4. We actually have defined a functor
Fun ( BG, C ∗ Cat ) → Fun ( G Orb , M ),which sends ˜ D to Hg G ˜ D , max . By construction this functor only depends on (cid:96) BG ( ˜ D ) andtherefore has a canonical factorization over (cid:96) BG : Fun ( BG, C ∗ Cat ) → Fun ( BG, C ∗ Cat ∞ ).Hence for ˜ D ∞ in Fun ( BG, C ∗ Cat ∞ ) we have a well-defined functorHg G ˜ D ∞ , max := Hg ∞ ( j G ! ( ˜ D ∞ )) : G Orb → M . This will be relevant for the statement of Proposition 17.5.The homotopy theoretic construction of Hg G ˜ D , max has the advantage that it is easy to derivesome of its formal properties. As an example, the next proposition states the compatibilityof the construction of Hg G ˜ D , max above with the induction along the inclusion of G into alarger group K . We have a commuting diagram of categories BG j G (cid:47) (cid:47) i (cid:15) (cid:15) G Orb i KG (cid:15) (cid:15) BK j K (cid:47) (cid:47) K Orb (17.6)where i : BG → BK is given by applying B to the inclusion of G into K , and i KG sendsthe G -orbit S to the K -orbit K × G S . For a functor E G : G Orb → M we let E G ( X )also denote the value of the corresponding M -valued equivariant homology theory on the G -topological space X . Furthermore, we let i KG, ! denote the left Kan extension functoralong i KG .Let ˜ D ∞ be in Fun ( BG, C ∗ Cat ∞ ) and Hg : C ∗ Cat → M be a functor to a cocompletetarget which sends unitary equivalences to equivalences. Proposition 17.5.
Assume that Hg preserves coproducts.1. We have an equivalence Hg Ki ! ˜ D ∞ , max (cid:39) i KG, ! Hg G ˜ D ∞ , max of functors from K Orb to M .2. For every K -topological space X we have Hg Ki ! ˜ D ∞ , max ( X ) (cid:39) Hg G ˜ D ∞ , max (Res KG ( X )) .Proof. We first show that Hg ∞ preserves coproducts. Then the claims of the propositionwill be consequences of general considerations that will be given in the appendix to thissection.The Dwyer–Kan localization (cid:96) : C ∗ Cat → C ∗ Cat ∞ of C ∗ Cat at the set of unitary equiv-alences is modeled by a combinatorial model category structure on C ∗ Cat in which allobjects are cofibrant (for details we refer to [Del12, Bun19]). Since in the latter all objectsof C ∗ Cat are cofibrant, for any family ( C i ) i ∈ I in C ∗ Cat the canonical morphism (cid:97) i ∈ I (cid:96) ( C i ) → (cid:96) (cid:0) (cid:97) i ∈ I C i (cid:1) ∞ preservescoproducts.We now turn to the actual proof of the proposition. Because Hg ∞ preserves coproducts,applying Lemma 17.32 with B = Hg ∞ and A = j G ! ˜ D ∞ we get i KG, ! Hg G ˜ D , max (cid:39) Hg ∞ ( i KG, ! j G ! ˜ D ∞ ) . We now use the commuting square (17.6) in order to rewrite the right-hand sideHg ∞ ( i KG, ! j G ! ˜ D ∞ ) (cid:39) Hg ∞ ( j K ! i ! ˜ D ∞ ) (cid:39) Hg Ki ! ˜ D ∞ , max . The concatenation of two equivalences gives the equivalence asserted in 1. Assertion 2 isnow an immediate consequence of Assertion 1 and Lemma 17.31.For an application of Proposition 17.5 see Example 17.26 below.
Remark 17.6.
We can apply the construction above to a unital C ∗ -algebra with G -actionin place of ˜ D . In the case of Hg = K C ∗ Cat one could try to compare this functor withthe functor constructed by Davis–L¨uck [DL98]. An immediate difference is that in theDavid–L¨uck case the value on
G/H is given by the K -theory of the reduced crossedproduct instead of the maximal one as in (17.5). The main goal of the present section isto construct a functor Hg G ˜ D ,r : G Orb → Sp whose values on orbits G/H are given byHg G ˜ D ,r (cid:39) Hg( ˜ D (cid:111) r H ) . We furthermore provide a comparison map c : Hg G ˜ D , max → Hg G ˜ D ,r which on the values is given by the canonical morphism between the maximal and reducedcrossed products. In particular, the restriction of c to G Fin
Orb is an equivalence.In order to incorporate the reduced crossed product we must work in a unital | G | -additive C ∗ -category ˜ C in Fun ( BG, C ∗ Cat ). We further consider an invariant closed subcategory˜ K and use for ˜ D the category ˜ K u in Fun ( BG, C ∗ Cat ) of unital objects in K (Definition2.10).We start by defining a functor C [ − ] : Fun ( BG,
Set ) → Fun ( BG, C ∗ Cat ) , (17.7)where Set is the small category of very small sets. We assume that C is idempotentcomplete. 117 efinition 17.7.
1. objects: For ˜ X in Fun ( BG,
Set ) we define C [ ˜ X ] in Fun ( BG, C ∗ Cat ) as follows:a) objects: The objects of C [ ˜ X ] are pairs ( C, ( p x ) x ∈ X ) of an object C of C and acommuting and mutually orthogonal family of projections p x in End C ( C ) suchthat C is isomorphic to the direct sum of their images (see Definition 4.15).b) morphisms: A morphism A : ( C, ( p x ) x ∈ X ) → ( C (cid:48) , ( p (cid:48) x ) x ∈ X ) in C [ ˜ X ] is a morphism A : C → C (cid:48) in C such that for all x, x (cid:48) we have p (cid:48) x Ap x = 0 unless x = x (cid:48) .c) composition and involution: These structures are inherited from C .d) The group G acts on C [ ˜ X ] by g ( C, ( p x ) x ∈ X ) := ( gC, ( gp g − x ) x ∈ X ) . The action of G on morphisms is inherited from ˜ C .2. morphisms: For a morphism f : ˜ X → ˜ X (cid:48) in Fun ( BG,
Set ) we define the functor C [ f ] : C [ ˜ X ] → C [ ˜ X (cid:48) ] as follows:a) objects: Let ( C, ( p x ) x ∈ X ) be an object of C [ ˜ X ] . Then C [ f ](( C, ( p x ) x ∈ X )) is theobject ( C, ( p x (cid:48) ) x (cid:48) ∈ X (cid:48) ) of C [ ˜ X (cid:48) ] , where p x (cid:48) := (cid:88) x ∈ f − ( { x (cid:48) } ) p x . (17.8) This object is well-defined by Lemma 4.13.b) morphisms: If A is a morphism from ( C, ( p x ) x ∈ X ) to ( C (cid:48) , ( p (cid:48) x ) x ∈ X ) in C [ ˜ X ] ,then C [ f ]( A ) is the same morphism A : C → C (cid:48) considered as a morphism from C [ f ](( C, ( p x ) x ∈ X )) to C [ f ](( C (cid:48) , ( p (cid:48) x (cid:48) ) x (cid:48) ∈ X (cid:48) )) in C [ ˜ X (cid:48) ] . One checks that C [ f ] is well-defined and compatible with the composition and involution.Idempotent completeness is used to construct C [ f ] since for the application of Lemma 4.13we must know that the projections (cid:80) x ∈ f − ( { x (cid:48) } ) p x are effective for all x (cid:48) in X (cid:48) .The support of an object ( C, ( p x ) x ∈ X ) of C [ ˜ X ] is defined by supp (( C, ( p x ) x ∈ X )) := { x ∈ X | p x (cid:54) = 0 } . xample 17.8. Let C be an object of C and y be a point in X . Then we consider theobject C x in C [ X ] given by ( C, ( p yx ) x ∈ X ), where p yx = 0 for all x in X except for x = y where p yy = id C . We say that C y is the object C placed at the point y in X . We have supp ( C y ) = { y } .Note that supp ( C [ f ](( C, ( p x ) x ∈ X ))) ⊆ f ( supp (( C, ( p x ) x ∈ X ))) . (17.9)For every ˜ X in Fun ( BG,
Set ) we define the G -invariant subcategory K C lf [ ˜ X ] of C [ ˜ X ] asfollows. Definition 17.9.
We define K C lf [ ˜ X ] in Fun ( BG, C ∗ Cat ) to be the full subcategory of C [ ˜ X ] of objects ( C, ( p x ) x ∈ X ) satisfying:1. For every x in X the image of p x is zero or isomorphic in C to an object of K u .2. supp (( C, ( p x ) x ∈ X )) is finite. It is clear that G preserves this subcategory.Let C be in C ∗ Cat nu and let D be a sub- C ∗ -category of C . Definition 17.10.
We say that D is hereditarily additive in C if every finite family ofobjects in D that admits an orthogonal sum in C also admits an orthogonal sum in D . Remark 17.11.
We often consider the case where ˜ K is an invariant ideal in ˜ C . In thiscase K u is automatically hereditarily additive in C . To see this, let ( C i ) i ∈ I be a finitefamily of objects of K u and assume that it admits an orthogonal sum ( C, ( e i ) i ∈ I ) in C .Because of e i = e i id C i and id C i ∈ K (since C i ∈ K u ) we can conclude that e i ∈ K forevery i in I . This implies that id C = (cid:80) i ∈ I e i e ∗ i ∈ K , hence C ∈ K u .On the other hand, we will consider the case where ˜ K is a single G -fixed unital object of˜ C , e.g., ˜ A in (cid:94) Hilb ( G )( ˜ A ). Except in degenerate cases K u will then not be hereditarilyadditive in C . Lemma 17.12. If K u is hereditarily additive in C , then Definition 17.9 describes asubfunctor K C lf [ − ] : Fun ( BG,
Set ) → Fun ( BG, C ∗ Cat ) of C [ − ] . roof. We must show that the subcategories are preserved by the functors C [ f ] formorphisms f : ˜ X → ˜ X (cid:48) in Fun ( BG,
Set ). Let ( C, ( p x ) x ∈ X ) be an object of K C lf [ ˜ X ]. Sincewe assume that K u is hereditarily additive in C we can conclude that for any finite subset F of X the image of the projection (cid:80) x ∈ F p x is isomorphic in C to an object of K u . Weapply this to the sums in (17.8) which are finite by Condition 17.9.2. Hence the images ofthe projections p x (cid:48) belong to K u for all x (cid:48) in X (cid:48) . Hence C [ f ] preserves Condition 17.9.1.Using (17.9) we furthermore see that C [ f ] preserves the Condition 17.9.2. Remark 17.13.
In [BE] we introduce categories of objects in C which are controlled by G -bornological coarse spaces. Assume that ˜ K is a G -invariant ideal in ˜ C . Then the functor K C [ − ] lf defined above (using Remark 17.11) corresponds to the functor ˜¯ C ctrlf (( − ) min , max )defined in [BE]. Remark 17.14.
We denote by
Fun ( BG,
Set ) i the wide subcategory of Fun ( BG,
Set ) ofmorphisms which are injective. If K u is not hereditarily additive in C , then K C lf [ − ] givesrise to a functor Fun ( BG,
Set ) i → Fun ( BG, C ∗ Cat ).Since we assume that C is | G | -additive, the C ∗ -category C [ ˜ X ] is also | G | -additive. Indeed,if ( C i , ( p i,x ) x ∈ X ) i ∈ I is a family of objects in C [ ˜ X ] with | I | ≤ | G | , then we can choose asum ( (cid:76) i ∈ I C i , ( e i ) i ∈ I ) in C . Using Proposition 4.3 one then checks that( (cid:77) i ∈ I C i , ( ⊕ i ∈ I p i,x ) x ∈ X , ( e i ) i ∈ I )represents the sum of this family in C [ ˜ X ]. If f : ˜ X → ˜ X is a morphism in Fun ( BG,
Set ),then C [ f ] preserves these sums. The functor C [ − ] from (17.7) therefore defines a functorfrom Fun ( BG,
Set ) to
Fun ( BG, (cid:92) C ∗ Cat | G | add ) (see Section 7 for notation). This allows usto define the reduced crossed product C [ ˜ X ] (cid:111) r G according to the Definition 10.8. Usingthe Lemma 10.10 we get a functor C [ − ] (cid:111) r G : Fun ( BG,
Set ) → C ∗ Cat . (17.10)For the next definition we assume that K u is hereditarily additive in C . Note that the setof objects of C [ ˜ X ] (cid:111) r G is the set of objects of C [ ˜ X ]. Definition 17.15.
We define the functor K C lf [ − ] (cid:111) r G : Fun ( BG,
Set ) → C ∗ Cat as the subfunctor of C [ − ] (cid:111) r G which sends ˜ X in Fun ( BG,
Set ) to the full subcategory K C lf [ ˜ X ] (cid:111) r G of C [ ˜ X ] (cid:111) r G on the objects of K C lf [ ˜ X ] . emark 17.16. Note that K C lf [ ˜ X ] (cid:111) r G is an abuse of notation. The reduced crossedproduct K C lf [ ˜ X ] (cid:111) r G is not defined intrinsically. It depends on the choice of the | G | -additive surrounding category C [ ˜ X ]. The more precise notation would be K C lf [ ˜ X ] (cid:111) C [ ˜ X ] r G (see Remark 10.11 and Lemma 10.12).The point of introducing the local finiteness condition by means of the subcategory ˜ K is toprevent the resulting category K C lf [ ˜ X ] (cid:111) r G from being countably additive, hence flasqueand therefore K -theoretically uninteresting (Example 9.4 and Proposition 11.12). Remark 17.17. If K u is not hereditarily additive in C , then K C lf [ − ] (cid:111) r G is still definedas a functor from Fun ( BG,
Set ) i to C ∗ Cat .We now calculate the value K C lf [ (cid:93) G/H ] (cid:111) r G for H a subgroup of G . Recall that ˜ C is Fun ( BG, C ∗ Cat ) and that ˜ K a closed, G -invariant subcategory of ˜ C . We assume that C is | G | -additive and idempotent complete. Note that we do not require that K u is additivein any sense. By abuse of notation we define the crossed product˜ K u (cid:111) r H := ˜ K u (cid:111) C r H , (17.11)(see Remark 10.11 for the decoration of the crossed product). Note that here and belowdo not write the symbol Res GH for better readability.Recall the notion of a Morita equivalence from Definition 14.6. Proposition 17.18.
We have a Morita equivalence ˜ K u (cid:111) r H → K C lf [ (cid:93) G/H ] (cid:111) r G .
Proof.
We first construct a functor i : ˜ C (cid:111) alg H → C [ (cid:93) G/H ] (cid:111) alg G as follows.1. objects: i sends the object C of C (cid:111) alg G (i.e., an object of C ) to the object C eH in C [ (cid:93) G/H ] (cid:111) alg G (i.e., an object of C [ (cid:93) G/H ]), see Example 17.8.2. morphisms: i sends a morphism ( f, h ) : C → C (cid:48) in ˜ C (cid:111) alg H with f : C → h − C (cid:48) (see [Bun, Defn. 5.1] or Page 62 for notation) to the morphism ( f, h ) : C eH → C (cid:48) eH in C [ (cid:93) G/H ] (cid:111) alg G . This is well-defined since h − ( C eH ) = ( h − C (cid:48) ) eH .The functor i is injective on objects and morphisms. It identifies ˜ C (cid:111) alg H with the fullsubcategory of C [ (cid:93) G/H ] (cid:111) alg G of objects which are supported on the class H in G/H .121e let D denote the full subcategory of C [ (cid:93) G/H ] (cid:111) alg G of objects which are supported ona single point of G/H . We claim that i : ˜ C (cid:111) alg H → D is a unitary equivalence. In order to construct an inverse (up to unitary isomorphism)functor q : D → ˜ C (cid:111) alg H (17.12)we choose a section t : G/H → G of the projection G → G/H such that t ( eH ) = e .1. objects: The functor q sends an object ( D, p D ) in D supported on (cid:96)H to the object q ( D, p D ) := t ( (cid:96)H ) − ( D, p D ) of ˜ C (cid:111) alg G . This object is supported on the class H and interpreted as an object t ( (cid:96)H ) − D of ˜ C (cid:111) alg H .2. morphisms: If ( f, g ) : ( D, p D ) → ( D (cid:48) , p D (cid:48) ) is a non-trivial morphism in D , then wehave f : ( D, p D ) → g − ( D (cid:48) , p D (cid:48) ) in C [ (cid:93) G/H ]. Assume that supp (( D, p D )) = (cid:96)H and supp (( D (cid:48) , p D (cid:48) )) = (cid:96) (cid:48) H . Then supp ( g − ( D (cid:48) , p D (cid:48) )) = g − (cid:96) (cid:48) H . Since f is non-trivial wehave g − (cid:96) (cid:48) H = (cid:96)H , and thus t ( (cid:96) (cid:48) H ) − gt ( (cid:96)H ) ∈ H . The functor q sends ( f, g ) to( t ( (cid:96)H ) − ( f ) , t ( (cid:96) (cid:48) H ) − gt ( (cid:96)H )) : t ( (cid:96)H ) − D → t ( (cid:96) (cid:48) H ) − D (cid:48) . We extend q by linearity. One checks compatibility with compositions and theinvolution in a straightforward manner.Then q ◦ i = id . In order to exhibit q as an inverse of i we furthermore construct a unitaryisomorphism u : i ◦ q → id . It is given by the family u = ( u ( D,p D ) ) ( D,p D ) ∈ Ob( D ) with u ( D,p D ) := ( id D , t ( (cid:96)H )) , where (cid:96)H is determined by supp ( D, p D ) = (cid:96)H . Indeed, u ( D,p D ) is a unitary isomorphismfrom t ( (cid:96)H ) − ( D, p D ) = i ( q ( D, p D )) to ( D, p D ). One checks naturality in a straightforwardmanner.Next we construct a functor j : L ( H, ˜ C ) → L ( G, C [ (cid:93) G/H ])in a similar manner, see Definition 10.1 for notation.1. objects: The functor j sends the object C in L ( H, ˜ C ) (i.e., an object of C ) to theobject C eH in L ( G, C [ (cid:93) G/H ]) (i.e., an object of C [ (cid:93) G/H ]).2. morphisms: Let f : C → C (cid:48) be a morphism in L ( H, ˜ C ). Then we have chosenorthogonal sums ( (cid:76) h ∈ H hC, ( e Hh ) h ∈ H ) in C and ( (cid:76) g ∈ G g ( C, p e ) , ( e Gg ) g ∈ G ) in C [ (cid:93) G/H ].The functor j sends f to j ( f ) := (cid:88) h (cid:48) ,h ∈ H e (cid:48) Gh (cid:48) i ( e (cid:48) ,H, ∗ h (cid:48) f e Hh , e ) e G, ∗ h . (17.13)122ne checks in a straightforward manner that the following square commutes:˜ C (cid:111) alg H i (cid:47) (cid:47) ρ C (cid:15) (cid:15) C [ (cid:93) G/H ] (cid:111) alg G ρ C [ (cid:94) G/H ] (cid:15) (cid:15) L ( H, ˜ C ) j (cid:47) (cid:47) L ( G, C [ (cid:93) G/H ])where the vertical functors are instances of (10.6). By an inspection of the definitionswe see that j is an isometric inclusion. Here we use that the orthogonal sums on bothsides are effectively constructed in C . While morphisms j ( C ) → j ( C (cid:48) ) are G × G -matricesof morphisms in C , the morphisms C → C (cid:48) in the domain of j are identified by j with H × H -submatrices representing morphisms between the same objects. The norm isinduced in both cases from the norm on the same morphism space in C .It follows from Definition 10.8 of the reduced crossed product that i extends by continuityto a functor ¯ i : ˜ C (cid:111) r H → C [ (cid:93) G/H ] (cid:111) r G .
The image of ¯ i is the full subcategory C [ (cid:93) G/H ] (cid:111) r G of objects which are supported on theclass H . We let ¯ D be the full subcategory of C [ (cid:93) G/H ] (cid:111) r G of objects supported on singlepoints. Then ¯ i : ˜ C (cid:111) r H → ¯ D is an equivalence. Indeed, the inverse q constructed above, see (17.12), extends by continuityto an inverse of ¯ i .We now consider the square˜ K u (cid:111) r H (cid:39) (1) (cid:47) (cid:47) (cid:15) (cid:15) K ¯ D k (cid:47) (cid:47) (cid:15) (cid:15) K C lf [ (cid:93) G/H ] (cid:111) r G (cid:15) (cid:15) ˜ C (cid:111) r H (cid:39) ¯ i (cid:47) (cid:47) ¯ D (2) (cid:47) (cid:47) C [ (cid:93) G/H ] (cid:111) r G (17.14)whose morphisms are defined as follows. The left vertical morphism is the inclusion of afull subcategory reflecting the definition (17.11) of ˜ K u (cid:111) r H . The right vertical morphismis induced by the embedding of a full subcategory K C lf [ (cid:93) G/H ] → C [ (cid:93) G/H ]. The category K ¯ D is the full subcategory of ¯ D of objects ( C, p C ) in ¯ D such that C belongs to K u . Thefunctor k is the inclusion. It is then clear that the restriction of ¯ i to ˜ K u (cid:111) r H takes valuesin K ¯ D . This provides the equivalence (1).It remains to show that the inclusion k : K ¯ D → K C lf [ (cid:93) G/H ] (cid:111) r G is a Morita equivalence. We will actually show the stronger statement that every object in K C lf [ (cid:93) G/H ] (cid:111) r G is isomorphic to a finite orthogonal sum of objects in K ¯ D . This implies123hat k ⊕ : ( K ¯ D ) ⊕ → ( K C lf [ (cid:93) G/H ] (cid:111) r G ) ⊕ (17.15)is already a unitary equivalence.Thus let ( C, p C ) be an object of K C lf [ (cid:93) G/H ] (cid:111) r G . By Definition 17.9 can we choose images( C ( gH ) , u gH ) of the projections p CgH for all gH in the finite set supp ( C, p C ) such that C ( gH ) belongs to K u . By Definition 17.7.1a we have an isomorphism (cid:88) gH ∈ supp ( C,p C ) u gH e ∗ gH : (cid:77) gH ∈ supp ( C,p C ) C ( gh ) → C in C . For any gH in supp ( C, p C ) we get an object ( C ( gh ) , p gH ) in K ¯ D , where p gH =( p gH(cid:96)H ) (cid:96)H ∈ G/H is such that p gH(cid:96)H = 0 if (cid:96)H (cid:54) = gH , and p gHgH = id C ( gH ) . We then have a unitaryisomorphism (cid:88) gH ∈ supp ( C,p C ) ( u gH e ∗ gH , e ) : (cid:77) gH ∈ supp ( C,p C ) ( C ( gh ) , p gH ) → ( C, p C )in K C lf [ (cid:93) G/H ] (cid:111) r G . Therefore ( C, p C ) is unitarily isomorphic to a finite orthogonal sumof objects of K ¯ D .Let Hg : C ∗ Cat nu → M be a functor. Recall that we are considering a unital | G | -additiveidempotent complete C ∗ -category ˜ C with strict G -action in Fun ( BG, C ∗ Cat ) with aninvariant closed subcategory ˜ K . We in addition assume that K u is hereditarily additive(Definition 17.10) in C . Note that G Orb is a full subcategory of
Fun ( BG,
Set ). Definition 17.19.
We define the functor Hg G ˜ K u ,r : G Orb → M as the composition Hg G ˜ K u ,r : G Orb → Fun ( BG,
Set ) K C lf [ − ] (cid:111) r G −−−−−−−→ C ∗ Cat Hg −→ M . (17.16)The next corollary of Proposition 17.18 justifies the notation announced in Remark 17.6. Corollary 17.20. If Hg is Morita invariant, then for every subgroup H of G we have anequivalence Hg( ˜ K u (cid:111) r H ) → Hg G ˜ K u ,r ( G/H ) . Let Hg : C ∗ Cat → M be a functor which sends unitary equivalences to equivalences, andlet F be a family of subgroups of G .In the case Hg = K C ∗ Cat we will use the more readable notation K G ˜ K u , max := (K C ∗ Cat ) G ˜ K u , max for the functor in (17.4), and K G ˜ K u ,r := (K C ∗ Cat ) G ˜ K u ,r for the functor in (17.16).124 roposition 17.21.
1. There exists a natural transformation c : Hg G ˜ K u , max → Hg G ˜ K u ,r .2. If Hg is Morita invariant and every member of F is amenable, then c | G F Orb : (Hg G ˜ K u , max ) | G F Orb → (Hg G ˜ K u ,r ) | G F Orb is an equivalence.3. If every member of F is K -amenable, then c | G F Orb : (K G ˜ K u , max ) | G F Orb → (K G ˜ K u ,r ) | G F Orb is an equivalence.
Before we start the proof of Proposition 17.21 we prove two intermediate lemmas.We let G (cid:48) be a second copy of G . Then we can form the functor φ : G Orb → Fun ( BG (cid:48) , Set )which sends S in G Orb to S considered as a G (cid:48) -set. We consider the set ˜ G in Fun ( BG × BG (cid:48) , Set ), where G (cid:48) -action is the right-action and the G -action is the left action on ˜ G .We let δ : Set → Spc denote the canonical map and recall the embedding of categories j G : BG → G Orb from (17.1).
Lemma 17.22.
We have an equivalence of functors ( j G × id BG (cid:48) ) ! δ BG × BG (cid:48) ( ˜ G ) (cid:39) δ G Orb × BG (cid:48) ( φ ) of functors from G Orb × BG (cid:48) to Spc .Proof.
The inverse map g (cid:55)→ g − induces an isomorphism ˜ G ∼ = → ( j G × id BG (cid:48) ) ∗ φ in Fun ( BG × BG (cid:48) , Set ). We get the morphism( j G × id BG (cid:48) ) ! δ BG × BG (cid:48) ( ˜ G ) ∼ = → ( j G × id BG (cid:48) ) ! ( j G × id BG (cid:48) ) ∗ δ G Orb × BG (cid:48) ( φ ) counit → δ G Orb × BG (cid:48) ( φ ) . We must show that the counit is an equivalence. To this end we evaluate it at
G/H in G Orb and get( j G × id BG (cid:48) ) ! ( j G × id BG (cid:48) ) ∗ δ BG × BG (cid:48) ( φ )( G/H ) (cid:39) colim ( G → G/H ) ∈ BG /G/H δ G Orb × BG (cid:48) ( φ )( G ) (cid:39) colim BH δ BG (cid:48) ( φ )( G ) (cid:39) δ BG (cid:48) ( φ )( G/H ) , where for the last equivalence we use that H acts freely on ˜ G and that therefore we cancalculate the colimit over BH before applying δ BG (cid:48) .125ince C ∗ Cat has all coproducts it is tensored over
Set . For D in C ∗ Cat the functor D ⊗ − : Set → C ∗ Cat is essentially uniquely determined by an isomorphism D ⊗ ∗ ∼ = D and the property that it preserves coproducts. If ˜ D is in Fun ( BG, C ∗ Cat ) and ˜ S is in Fun ( BG,
Set ), then we can consider ˜ D ⊗ ˜ S in Fun ( BG, C ∗ Cat ).Similarly, the ∞ -category C ∗ Cat ∞ is cocomplete and hence tensored over Spc . For D ∞ in C ∗ Cat ∞ the functor D ∞ ⊗ − : Spc → C ∗ Cat ∞ is essentially uniquely determinedby an equivalence D ∞ ⊗ ∗ (cid:39) D ∞ and the property that it preserves colimits. If ˜ D ∞ is in Fun ( BG, C ∗ Cat ∞ ) and if ˜ X is Fun ( BG,
Spc ), then we can consider ˜ D ∞ ⊗ ˜ X in Fun ( BG, C ∗ Cat ∞ ).The functor δ : Set → Spc preserves coproducts. Since also the localization (cid:96) : C ∗ Cat → C ∗ Cat ∞ preserves coproducts, for S in Set we have an equivalence (cid:96) ( D ⊗ S ) (cid:39) (cid:96) ( D ) ⊗ δ ( S )and similarly, (cid:96) BG ( ˜ D ⊗ ˜ S ) (cid:39) (cid:96) BG ( ˜ D ) ⊗ δ BG ( ˜ S ) (17.17)for all ˜ D in Fun ( BG, C ∗ Cat ) and ˜ S in Fun ( BG,
Set ).Let ˜ D be in Fun ( BG, C ∗ Cat ). We write ˜ D (cid:48) in Fun ( BG (cid:48) , C ∗ Cat ) for ˜ D considered withthe G (cid:48) -action. Lemma 17.23.
We have an equivalence j G ! ( (cid:96) BG ( ˜ D )) (cid:39) (cid:96) G Orb (( ˜ D (cid:48) ⊗ φ ) (cid:111) G (cid:48) ) .Proof. We have the equivalence (cid:96) BG (cid:48) ( ˜ D (cid:48) ) ⊗ δ BG × BG (cid:48) ( ˜ G ) (cid:39) (cid:96) BG × BG (cid:48) ( ˜ D (cid:48) ⊗ ˜ G ) (cid:39) (cid:96) BG × BG (cid:48) ( ˜ D ⊗ ˜ G ) (cid:39) (cid:96) BG ( ˜ D ) ⊗ δ BG × BG (cid:48) ( ˜ G ) , (17.18)where the middle equivalence is given by ( C, h ) (cid:55)→ ( hC, h ). It sends the previous diagonalaction of G (cid:48) to the right action of G (cid:48) on ˜ G , and the previous left action of G on ˜ G to thediagonal action. We have colim BG (cid:48) δ BG × BG (cid:48) ( ˜ G ) (cid:39) ∗ since G (cid:48) acts freely on ˜ G so that we can calculate the colimit before going from sets tospaces. Applying colim BG (cid:48) to (17.18) we get the equivalence colim BG (cid:48) ( (cid:96) BG (cid:48) ( ˜ D (cid:48) ) ⊗ δ BG × BG (cid:48) ( ˜ G )) (cid:39) (cid:96) BG ( ˜ D ) . We now apply j G ! and use that this left Kan extension functor preserves colimits to get colim BG (cid:48) ( j G × id BG (cid:48) ) ! ( (cid:96) BG (cid:48) ( ˜ D (cid:48) ) ⊗ δ BG × BG (cid:48) ( ˜ G )) (cid:39) j G ! colim BG (cid:48) ( (cid:96) BG ( ˜ D (cid:48) ) ⊗ δ BG × BG (cid:48) ( ˜ G )) (cid:39) j G ! ( (cid:96) BG ( ˜ D )) . (17.19)Finally, using Lemma 17.22, Equation (17.17), and that (cid:96) BG ( ˜ D (cid:48) ) ⊗ − preserves colimits126e can rewrite the domain of (17.19) as colim BG (cid:48) ( j G × id BG (cid:48) ) ! ( (cid:96) BG ( ˜ D (cid:48) ) ⊗ δ BG × BG (cid:48) ( ˜ G )) (cid:39) colim BG (cid:48) ( (cid:96) BG (cid:48) ( ˜ D (cid:48) ) ⊗ δ G Orb × BG (cid:48) ( φ )) (cid:39) colim BG (cid:48) (cid:96) G Orb × BG (cid:48) ( ˜ D (cid:48) ⊗ φ ) (cid:39) (cid:96) G Orb (( ˜ D (cid:48) ⊗ φ ) (cid:111) G (cid:48) ) , where for the last equivalence we use [Bun, Thm. 7.8] Proof of Proposition 17.21.
We now come back to our original situation and specialize theabove results to ˜ D := ˜ K u . We define a transformation ν : ˜ K (cid:48) ,u ⊗ φ → K C lf [ φ ( − )]of functors from G Orb to Fun ( BG (cid:48) , C ∗ Cat ). Note that for T in G Orb an object of˜ K (cid:48) ,u ⊗ φ ( T ) is given by a pair ( C, t ) of an object C of K u and a point t in T .1. objects: The evaluation ν T of ν at T sends the object ( C, t ) in ˜ K (cid:48) ,u ⊗ φ ( T ) to theobject C t in K C lf [ T ]. Recall that C t is the object C placed at the point t (Example17.8).2. morphisms: A morphism ( C, t ) → ( C (cid:48) , t (cid:48) ) in ˜ K (cid:48) ,u ⊗ φ ( T ) only exists if t = t (cid:48) . In thiscase it is given by a morphism f : C → C (cid:48) in K u . The evaluation ν T of ν at T sendsthis morphism to the morphism f t : C t → C (cid:48) t .One checks that ν T is a well-defined functor between C ∗ -categories and G (cid:48) -equivariant.Furthermore, the family ν = ( ν T ) T ∈ G Orb is a natural transformation. We get an inducedtransformation ν (cid:111) G : ( ˜ K (cid:48) ,u ⊗ φ ) (cid:111) G (cid:48) → K C lf [ φ ( − )] (cid:111) G (cid:48) ! → K C lf [ φ ( − )] (cid:111) r G (cid:48) ∼ = K C lf [ − ] (cid:111) r G , (17.20)where the marked natural transformation is the transformation from the maximal to thereduced crossed product. We furthermore apply Hg ∞ ◦ (cid:96) G Orb and get the transformation c : Hg G ˜ K u , max Def.17 . (cid:39) Hg ∞ ( j G ! ( (cid:96) BG ( ˜ K u ))) Lem.17 . (cid:39) Hg ∞ ( (cid:96) G Orb (( ˜ K (cid:48) ,u ⊗ φ ) (cid:111) G (cid:48) )) Hg ∞ ( (cid:96) G Orb ((17.20))) → Hg ∞ ( (cid:96) G Orb ( K C lf [ − ] (cid:111) r G (cid:48) )) (17.3) (cid:39) Hg( K C lf [ − ] (cid:111) r G ) Def.17 . (cid:39) Hg G K u ,r . This finishes the construction of the morphism c in Assertion 1.To see Assertions 2 and 3 note that the evaluation of c at G/H reduces under theequivalences in 17.20 and (17.5) to the canonical morphismHg( q ˜ K u ) : Hg( ˜ K u (cid:111) H ) → Hg( ˜ K u (cid:111) r H )127rom the maximal to the reduced crossed product by H . If H is amenable, then it is anequivalence by Theorem 10.19. If H is K -amenable, then in the special case of Hg = K C ∗ Cat it is an equivalence by Theorem 12.12.We now relate the functor K DL ,G C : G Orb → Sp introduced by Davis–L¨uck in [DL98]with the constructions of the present paper. We will actually consider its straightforwardgeneralization to the case of a unital C ∗ -algebra A in place of C and a functor Hg : C ∗ Cat → M which sends unitary equivalences to equivalences and is Morita invariant in place ofK C ∗ Cat . We then get a functor Hg DL ,GA : G Orb → M whose value on the orbit G/H is given byHg DL ,GA ( G/H ) (cid:39) Hg( A (cid:111) r H ) . (17.21)Here we consider A with the trivial G -action. Associated to A we consider the C ∗ -category (cid:94) Hilb ( A ) in Fun ( BG, C ∗ Cat ) with the trivial G -action and with the invariant ideal ˜ K ( A )of compact operators (Example 16.13). Proposition 17.24.
There is a canonical equivalence of functors Hg DL ,GA (cid:39) Hg G ˜ K ( A ) u ,r from G Orb to M .Proof. Let
Groupoids inj denote the category of very small groupoids and faithful mor-phisms. We start with explaining the construction of Hg DL ,GA following [DL98]. We havea functor Fun ( BG,
Set ) → Groupoids inj which sends S in Fun ( BG,
Set ) to the actiongroupoid S (cid:120) G . The latter has the following description:1. objects: The set of objects of S (cid:120) G is the set S .2. morphisms: For s, s (cid:48) in S the set of morphisms from s to s (cid:48) is the subset { g ∈ G | gs = s (cid:48) } of G .3. The composition is inherited from the multiplication in G .A morphism f : S → S (cid:48) in Fun ( BG,
Set ) induces a morphism f (cid:120) G : S (cid:120) G → S (cid:48) (cid:120) G in Groupoids inj which sends s in S to f ( s ) in s (cid:48) and acts as natural inclusions on morphismsets. 128e have a functor C ∗ A,r : Groupoids inj → C ∗ Cat defined in [DL98] as follows. For a groupoid S we first form the algebraic tensor product A ⊗ alg S in ∗ Cat C as in [Bun19, Sec. 6]. Its objects are the objects of S . But instead ofcompleting in the maximal norm (which would give A ⊗ max S ) we complete in the reducednorm described in [DL98, Sec. 6]. To do this, for any two objects s, s (cid:48) in S we canonicallyembed Hom A ⊗ alg S ( s, s (cid:48) ) into the adjointable bounded operators between A -Hilbert C ∗ -modules B ( L ( Hom S ( s , s ) , A ) , L ( Hom S ( s , s (cid:48) ) , A )) and take the supremum of the normsof the images over all choices if s in S . We let C ∗ A,r ( S ) be the completion of A ⊗ alg S .A morphism f : S → S (cid:48) in Groupoids inj induces a morphism C ∗ A,r ( S ) → C ∗ A,r ( S (cid:48) ) in thenatural way. At this point it is important that we only consider faithful morphisms betweengroupoids. The functor C ∗ A,r extends to a functor between 2-categories (of groupoids,faithful morphisms and equivalences on the one hand; and C ∗ -categories, functors andunitary equivalences on the other hand) and sends equivalences of groupoids to unitaryequivalences of C ∗ -categories.The functor Hg DL ,GA is then defined byHg DL ,GA : G Orb S (cid:55)→ S (cid:120) G −−−−−→ Groupoids inj C ∗ A,r −−−→ C ∗ Cat Hg −→ M . If H is a subgroup of G , then we have an equivalence of groupoids ( ∗ (cid:120) H ) (cid:39) → (( G/H ) (cid:120) G ) which sends ∗ to the class H . This equivalence induces a unitary equivalence A (cid:111) r H ∼ = C ∗ A,r ( ∗ (cid:120) H ) (cid:39) → C ∗ A,r (( G/H ) (cid:120) G ) (17.22)in C ∗ Cat which yields (17.21) by applying Hg.We now define a natural transformation of functors κ : C ∗ A,r ( − (cid:120) G ) → ˜ K ( A ) (cid:94) Hilb ( A ) lf [ (cid:101) − ] (cid:111) r G . (17.23)The evaluation of κ S of κ at S in G Orb is given as follows:1. objects: κ S sends the object s in S = Ob( C ∗ A,r ( S (cid:120) G )) to the object A s in ˜ K ( A ) (cid:94) Hilb ( A ) lf [ (cid:101) S ] (cid:111) r G (see Example 17.8), where we consider A as an object of˜ K ( A ) u (note that A is unital by assumption).2. morphisms: Let s, s (cid:48) be in S , let g in G be such that gs = s (cid:48) , and let a be in A .Then we can consider ( a, g ) as a morphism in A ⊗ alg ( S (cid:120) G ), and therefore as amorphism in A ⊗ r ( S (cid:120) G ). We can consider the right-multiplication by a as amorphism a : A s → A s (cid:48) = gA s in ˜ K ( A ) (cid:94) Hilb ( A ) lf [ (cid:101) S ]. The functor κ S sends ( a, g ) tothe morphism ( a, g ) : A s → A s (cid:48) in ˜ K ( A ) (cid:94) Hilb ( A ) lf [ (cid:101) S ] (cid:111) r G .We extend κ S by linearity and continuity.129ne checks that κ S is well-defined and that the family κ := ( κ S ) S ∈ G Orb is a naturaltransformation. In order to check that κ S extends by continuity we do not have toconsider estimates. We just check that for a subgroup H of G the functor κ G/H identifies C ∗ A,r (( G/H ) (cid:120) G ) with the subcategory A ¯ D of A (cid:94) Hilb ( A ) lf [ (cid:101) S ] (cid:111) r G appearing in (17.14).This follows from the fact that both receive unitary equivalences from A (cid:111) r H by (17.22)and the map (1) in (17.14).We consider A as a G -invariant one-object subcategory of ˜ K ( A ) u . Let H be a subgroup of G . The inclusion induces a functor A (cid:94) Hilb ( A ) lf [ G/H ] (cid:111) r G → K ( A ) u (cid:94) Hilb ( A ) lf [ G/H ] (cid:111) r G . (17.24)Note that A is not hereditally additive in (cid:94) Hilb ( A ) so that we do not have functoriality inthe argument G/H . The functor (17.24) in turn induces the morphism in the statementbelow by applying Hg.The following lemma is an essential step in the proof of Proposition 17.24 but might beinteresting in its own right.Recall that Hg : C ∗ Cat → M is a functor which sends unitary equivalences to equivalencesand is Morita invariant. Lemma 17.25.
We have an equivalence Hg GA,r ( G/H ) → Hg G ˜ K ( A ) u ,r ( G/H ) . (17.25) Proof.
Under Corollary 17.20 the morphism (17.25) corresponds toHg( A (cid:111) (cid:93) Hilb ( A ) r H ) → Hg( ˜ K ( A ) u (cid:111) (cid:93) Hilb ( A ) r H ) (17.26)induced by the inclusion the inclusion A → ˜ K ( A ) u . As observed in Example 16.13 wehave an equality K ( A ) u ∼ = Hilb ( A ) fg , proj . The inclusion A → Hilb G ( A ) fg , proj is a Moritaequivalence by Example 14.7. By Proposition 14.8 we conclude that A (cid:111) (cid:93) Hilb ( A ) r H → ˜ K ( A ) u (cid:111) (cid:93) Hilb ( A ) r H is a Morita equivalence and hence (17.26) is an equivalence.We finish now the proof of Proposition 17.24. The evaluation of Hg( κ ), where κ is as in13017.23), at G/H has the following factorization:Hg DL ,GA ( G/H ) Def. (cid:39)
Hg( C ∗ A,r (( G/H ) (cid:120) G )) Hg( κ G/H ) , (17.23) (cid:39) Hg( A ¯ D ) Hg( k ) , (17.14) (cid:39) Hg( A (cid:94) Hilb ( A ) lf [ (cid:93) G/H ] (cid:111) r G ) Lem.17 . (cid:39) Hg( ˜ K ( A ) u (cid:94) Hilb ( A ) lf [ (cid:93) G/H ] (cid:111) r G ) Def. (cid:39) Hg G ˜ K ( A ) u ( G/H )through equivalences.
Example 17.26.
Let ˜ C in Fun ( BG, C ∗ Cat ) be a | G | -additive idempotent complete C ∗ -category with G -action with an invariant ideal ˜ K . Assume that we are also given aninclusion of G as a subgroup into a larger group K . As before we let i : BG → BK be theinduced functor. Finally assume that we have chosen an | K | -additive idempotent completeobject Ind KG ( ˜ C ) in Fun ( BK, C ∗ Cat ) with an invariant ideal
Ind KG ( ˜ K ) such that there isan equivalence i ! (cid:96) BG ( ˜ K u ) (cid:39) (cid:96) BK ( Ind KG ( ˜ K u )).Let Hg : C ∗ Cat → M be a functor which sends unitary equivalences to equivalences andis Morita invariant. Corollary 17.27.
1. If M is cocomplete and Hg preserves coproducts, then for every K -CW-complex X whose stabilizers are amenable we have an equivalence Hg K Ind KG ( ˜ K u ) ,r ( X ) (cid:39) Hg G ˜ K u ,r (Res KG ( X )) . (17.27)
2. For every K -CW-complex X whose stabilizers are K -amenable we have an equivalence K K Ind KG ( ˜ K u ) ,r ( X ) (cid:39) K G ˜ K u ,r (Res KG ( X )) . (17.28) Proof.
We let Am and K-Am denote the families of amenable and K -amenable subgroups.The presheaves Y K ( X ) and Y G (Res KG ( X )) (see (17.34)) are supported on K Am Orb and G Am Orb , respectively (or on K K-Am
Orb , resp. G K-Am
Orb in the second case). In viewof (17.36) and Proposition 17.21.2 we have equivalencesHg K Ind KG ( ˜ K u ) , max ( X ) (cid:39) Hg K Ind KG ( ˜ K u ) ,r ( X ) (17.29)and Hg G ˜ K u , max (Res KG ( X )) (cid:39) Hg G ˜ K u ,r (Res KG ( X )) , (17.30)131y Proposition 17.5.2 we have an equivalenceHg K Ind KG ( ˜ K u ) , max ( X ) (cid:39) Hg G ˜ K u , max (Res KG ( X ) . The combination of these equivalences yields the equivalence (17.27). For (17.28) weuse Proposition 17.21.3 to conclude the equivalences (17.29) and (17.30) in the case ofHg = K C ∗ Cat (note that K C ∗ Cat preserves coproducts by Corollary 14.15).We apply the corollary to X = E Fin K . Since Res KG ( E Fin K ) (cid:39) E Fin G we get an equivalenceHg K Ind KG ( ˜ K u ) ,r ( E Fin K ) (cid:39) Hg G ˜ K u ,r ( E Fin G ) . (17.31)In the case of Hg = K C ∗ Cat the left and right hand sides of this equivalence constitute thedomains of corresponding Baum–Connes assembly maps. In this case such an equivalence(with a completely different model of equivariant K -homology and a completely differentproof) has first been obtained by [OO97], see [CE01, Thm. 2.2]. The comparison of modelswill be discussed further in [BEL]. Remark 17.28.
The following Theorem 17.29 is one of the main motivations for thefollow-up paper [BE] for which the present paper provides the foundations concerning C ∗ -categories. Let ˜ C be in Fun ( BG, C ∗ Cat ) and ˜ K be an invariant closed subcategory.We consider the case Hg = K C ∗ Cat in which case we will use the more readable notationK G ˜ K u ,r := (K C ∗ Cat ) G ˜ K u ,r for the functor defined in Definition 17.19. Recall the notion of aCP-functor from [BEKW20c]. Theorem 17.29 ([BE]) . Assume:1. ˜ K is an invariant ideal in ˜ C .2. ˜ C is idempotent complete and admits all very small sums.Then K G ˜ K u ,r : G Orb → Sp is a CP-functor. As explained in [BEKW20c], [BE, Sec. 1] and in [BCKW, Sec. 6.5] being a CP-functor hasinteresting consequences for the injectivity of assembly maps involving this functor.
Remark 17.30.
In this remark we explain why there can not be a simple argument(like in [BCKW, Thm. 6.25] or [BEKW20c, Ex. 2.6]) showing under the conditions of theTheorem 17.29 that K G ˜ K u ,r is a hereditary CP-functor.132e recall the notion of a hereditary CP-functor from [BEKW20c, Def. 2.5]. If φ : G → Q isa surjective homomorphism of groups, then we have a functor Res φ : Q Orb → G Orb whichsends a Q -orbit to the same underlying set with the G -action induced via φ . By [BEKW20c,Def. 2.5], a functor M : G Orb → M is a herditary CP-functor if M ◦ Res φ : Q Orb → M is a CP-functor for every surjective homomorphism φ : G → Q .In [BCKW, Thm. 6.25] we have seen that the construction of CP-functors like in Definition17.4 by a left-Kan extension along j G : BG → G Orb followed by application of Hg easilyimplies that the resulting functors are in fact also hereditary. In the following we explainwhy this does not directly extend to the case of C ∗ -categories.Let ˜ D be in Fun ( BG, C ∗ Cat ) and consider the functor Hg G ˜ D , max : G Orb → M from (17.3).Let φ : G → Q be a surjective homomorphism. Then we can choose ˜ E in Fun ( BQ, C ∗ Cat )such that Bφ ! (cid:96) BG ( ˜ D ) (cid:39) (cid:96) BQ ( ˜ E ). By the same proof as in [BCKW, Thm. 6.25] we get anequivalence Hg G ˜ D , max ◦ Res φ (cid:39) Hg Q ˜ E , max . (17.32)We now specialize to Hg := K C ∗ Cat and ˜ D := ˜ K u , where ˜ K is a G -invariant ideal in some˜ C in Fun ( BG, C ∗ Cat ) which is idempotent complete and admits small orthogonal sums.As in [BCKW] we would like to say that K Q ˜ E , max is a CP-functor by Theorem 17.29 andconclude that K G ˜ D , max ◦ Res φ is a CP-functor, too. The coarse geometry approach of [BE]dictates to work with the reduced versions of these functors on the orbit category, andthat the coefficient category has a special form.In order to solve the first problem we must ensure that the comparison morphismsK G ˜ D , max ◦ Res φ → K G ˜ D ,r ◦ Res φ , K Q ˜ E , max → K Q ˜ E ,r (17.33)are equivalences. For this it suffices to assume that all subgroups of G containing ker ( φ )and all subgroups of Q are K -amenable (by a similar argument as for Proposition 17.21.3).This includes the assumption that G itself is K -amenable (implying that all its subgroupsare K -amenable) which is a very restrictive assumption for the application to the proof ofsplit injectivity of the Baum–Connes assembly map.In addition, we do not have any nice condition ensuring ˜ E can be taken to be of the specialform ˜ L u for some invariant ideal L in an idempotent and small, sum complete C ∗ -categorywith Q -action. Let K be a group and K Top be the category of K -topological spaces. A morphism f : X → X (cid:48) in K Top is an equivariant weak equivalence if it induces weak equivalences133etween the fixed-points sets f H : X H → X (cid:48) ,H for all subgroups H of K . In the following let Map K Top ( − , − ) denote the topological mapping space of equivariant maps and (cid:96) : Top → Spc be the canonical morphism which presents the ∞ -category Spc as the Dwyer–Kanlocalization of
Top at the weak equivalences. By Elmendorf’s theorem the functor Y K : K Top → PSh ( K Orb ) , X (cid:55)→ ( S (cid:55)→ (cid:96) ( Map K Top ( S disc , X ))) (17.34)presents the localization K Top [ W − K ] of K Top at the class of equivariant weak equivalences W K . Here S disc denotes the K -orbit S considered as discrete K -topological space.For a subgroup G of K we have an adjunction Ind KG : G Top (cid:28) K Top : Res KG , where the induction functor is given by X (cid:55)→ Ind KG ( X ) := K × G X .
Considering the orbit category K Orb as a full subcategory of K Top of discrete transitive K -topological spaces, the induction functor restricts to the functor i KG : G Orb → K Orb . It is a formal consequence of the definitions that K Top
Res KG (cid:47) (cid:47) Y K (cid:15) (cid:15) G Top Y G (cid:15) (cid:15) PSh ( K Orb ) i K, ∗ G (cid:47) (cid:47) PSh ( G Orb ) (17.35)commutes. A functor E G : G Orb → M with cocomplete target represents an M -valued G -equivariant homology theory denoted by the same symbol E : G Top → M . We formthe left Kan extension G Orb E G (cid:47) (cid:47) Yoneda (cid:39) (cid:39) ⇒ M . PSh ( G Orb ) ˆ E G (cid:56) (cid:56) Then the value of the homology theory on X in K Top is given by E G ( X ) (cid:39) ˆ E G ( Y G ( X )) . (17.36)We form the left Kan extension E K := i KG, ! E : K Orb → M of E G as in G Orb E G (cid:47) (cid:47) i KG (cid:37) (cid:37) ⇒ M .K Orb E K (cid:58) (cid:58) It represents a K -equivariant homology theory. Let X be in K Top .134 emma 17.31.
We have a natural equivalence E K ( X ) (cid:39) E G (Res KG ( X )) .Proof. We have E G (Res KG ( X )) (cid:39) ˆ E G ( Y G (Res KG ( X ))) (17.35) (cid:39) ˆ E G ( i K, ∗ G ( Y K ( X ))) . Let y K Orb : K Orb → PSh ( K Orb ) denote the Yoneda embedding. Then we have anequivalence E K (cid:39) i KG, ! E G (cid:39) ˆ E G ◦ i K, ∗ G ◦ y K Orb which implies ˆ E K (cid:39) ˆ E G ◦ i K, ∗ G . We getˆ E G ( i K, ∗ G ( Y K ( X ))) (cid:39) ˆ E K ( Y K ( X )) (cid:39) E K ( X ) . The desired equivalence follows from concatenating the two displayed chains of equivalences.The left Kan extension functor i KG, ! only involves forming coproducts. More precisely, wehave the following assertion. Let A : G Orb → A be a functor with a cocomplete targetand B : A → B be a second functor to a cocomplete target B . Lemma 17.32. If B preserves coproducts, then the canonical transformation is an equiv-alence i KG, ! ( B ◦ A ) (cid:39) B ◦ i KG, ! A .Proof. We have a natural transformation i KG, ! ( B ◦ A ) → B ◦ i KG, ! A . We use the pointwiseformula for the left Kan extension in order to evaluate this transformation at S in K Orb .The objects of G Orb /S are morphisms K × G T → S for T in G Orb which are in bijectionwith morphisms T → Res KG ( S ) in Fun ( BG,
Set ). Hence the category G Orb /S decomposesinto a union of categories G Orb /R , where R runs over the set G \ S of G -orbits in Res GK ( S ).Each component has a final object R . Hence we get the following chain of equivalences:( i KG, ! ( B ◦ A ))( S ) (cid:39) (cid:97) R ∈ G \ S B ( A ( R )) (cid:39) B (cid:0) (cid:97) R ∈ G \ S A ( R ) (cid:1) (cid:39) B (( i KG, ! A )( S )) (cid:39) ( B ◦ i KG, ! A )( S ) . References [AV] J. Antoun and Ch. Voigt. On bicolimits of C ∗ -categories. arXiv:2006.06232.[BCKW] U. Bunke, D.-Ch. Cisinski, D. Kasprowski, and Ch. Winges. Controlled ob-jects in left-exact ∞ -categories and the Novikov conjecture. arXiv:1911.02338.135BE] U. Bunke and A. Engel. Topological equivariant coarse K -homology. Inpreparation.[BE20] U. Bunke and A. Engel. Homotopy theory with bornological coarse spaces ,volume 2269 of
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