A stable \infty-category for equivariant KK-theory
AA stable ∞ -category for equivariant KK -theory Ulrich Bunke ∗ Alexander Engel † Markus Land ‡ March 1, 2021
Abstract
For a countable group G we construct a small, idempotent complete, symmetricmonoidal, stable ∞ -category KK G sep whose homotopy category recovers the trian-gulated equivariant Kasparov category of separable G - C ∗ -algebras, and exhibit itsuniversal property. Likewise, we consider an associated presentably symmetricmonoidal, stable ∞ -category KK G which receives a symmetric monoidal functor kk G from possibly non-separable G - C ∗ -algebras and discuss its universal property. Inaddition to the symmetric monoidal structures, we construct various change-of-groupfunctors relating these KK-categories for varying G . We use this to define and es-tablish key properties of a (spectrum valued) equivariant, locally finite K -homologytheory on proper and locally compact G -topological spaces, allowing for coefficientsin arbitrary G - C ∗ -algebras. Finally, we extend the functor kk G from G - C ∗ -algebrasto G - C ∗ -categories. These constructions are key in a companion paper about a formof equivariant Paschke duality and assembly maps. Contents
1. Introduction and statements 2 ∞ -category of separable G - C ∗ -algebras . . . . . . . . . . . . 21.2. A presentable stable ∞ -category for G - C ∗ -algebras . . . . . . . . . . . . . 61.3. Equivariant, locally finite K -homology . . . . . . . . . . . . . . . . . . . . 71.4. The s-finitary extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ∗ Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, [email protected] † Mathematisches Institut, Westf¨alische Wilhelms–Universit¨at M¨unster, 48149 M¨unster, [email protected] ‡ Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, [email protected] a r X i v : . [ m a t h . OA ] F e b .5. Change of groups functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6. Extensions to C ∗ -categories . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2. A stable ∞ -category of separable G - C ∗ -algebras 193. The s-finitary extension 374. Change of groups functors 465. Locally finite K -homology 636. Extension to C ∗ -categories 697. Tensor products of C ∗ -categories 83A. Applications to assembly maps 99References 103Acknowledgements. Ulrich Bunke was supported by the SFB 1085 (Higher Invariants)funded by the Deutsche Forschungsgemeinschaft (DFG).Alexander Engel acknowledges financial support by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) through the Priority Programme SPP 2026 “Ge-ometry at Infinity” (EN 1163/5-1, project number 441426261, Macroscopic invariants ofmanifolds) and through Germany’s Excellence Strategy EXC 2044-390685587, MathematicsM¨unster: Dynamics – Geometry – Structure.Markus Land was supported by the research fellowship DFG 424239956, and by the DanishNational Research Foundation through the Copenhagen Centre for Geometry and Topology(DNRF151).
1. Introduction and statements ∞ -category of separable G - C ∗ -algebras Let G be a countable group and A , B be G - C ∗ -algebras, i.e. C ∗ -algebras with an actionof G by automorphisms. In this situation, we have the abelian group KK G ( A, B ) intro-duced by Kasparov in his work on the Novikov conjecture [Kas88]. This group dependscontravariantly on the first algebra A and covariantly on the second algebra B .The construction of KK G ( A, B ) can be generalized to graded G - C ∗ -algebras, to families2f C ∗ -algebras parametrized by a space as in [Kas88], or to the case where G is a locallycompact groupoid [LG99]. Though we think that many of our constructions also workin more general situations, in the present paper we will stick to the classical situationsince this is what immediately generalizes to C ∗ -categories, and what we need for theapplications to Paschke duality and assembly maps in [BEL].If we restrict to separable G - C ∗ -algebras, then the Kasparov product KK G ( A, B ) ⊗ KK G ( B, C ) → KK G ( A, C ) , which has also been introduced in [Kas88], can be considered as the composition law of an Ab -enriched category KK G . The objects of this category are the separable G - C ∗ -algebrasand the morphism groups are given by Hom KK G ( A, B ) := KK G ( A, B ) . The category KK G is additive and the sum is represented by the direct sum of C ∗ -algebras.As observed and exploited in [MN06], KK G has a canonical refinement to a triangulatedcategory.In the present paper, we will use the language of ∞ -categories. Standard references are[Lur09], [Cis19]. Of particular importance to us are stable ∞ -categories as developed in[Lur, Ch. 1]. The homotopy category of any stable ∞ -category C is canonically triangulated[Lur, Sec. 1.1.2], and many triangulated categories arise in this way. The first objectiveof this paper is to show that the triangulated categories KK G are no exception to thisprinciple.The first result of this paper is the construction of a stable ∞ -category KK G sep whosehomotopy category is canonically equivalent, as a triangulated category, to KK G . Thisgeneralizes a construction of [LN18] from the non-equivariant to the equivariant case. Inthe following we provide the precise statement.Let Fun ( BG, C ∗ Alg nusep ) denote the category of separable, possibly non-unital C ∗ -algebraswith G -action. We then have a canonical functorkk G : Fun ( BG, C ∗ Alg nusep ) → KK G (1.1)which is the identity on objects and sends f : A → B to the element [ f ] in KK G ( A, B )represented by the Kasparov (
A, B )-module (
B, f,
Definition 1.1.
A morphism f in Fun ( BG, C ∗ Alg nusep ) is called a kk G -equivalence if kk G ( f ) is an isomorphism. We let W kk G denote the collection of all kk G -equivalences. The following definition is thedirect generalization of [LN18] to the equivariant case.3 efinition 1.2. We define the ∞ -category KK G sep := Fun ( BG, C ∗ Alg nusep )[ W − G ] and let kk G sep : Fun ( BG, C ∗ Alg nusep ) → KK G sep denote the canonical functor. Here the ∞ -category Fun ( BG, C ∗ Alg nusep )[ W − G ] denotes the Dwyer–Kan localization of Fun ( BG, C ∗ Alg nusep ) at the collection W kk G of the kk G -equivalences. Such a Dwyer–Kanlocalization exists and is characterized by a universal property, see Remark 2.5 for theprecise statement. We have the following theorem: Theorem 1.3.
1. The ∞ -category KK G sep is stable.2. We have a canonical factorization Fun ( BG, C ∗ Alg nusep ) kk G (cid:47) (cid:47) kk G sep (cid:40) (cid:40) KK G KK G sep ho (cid:59) (cid:59) (1.2) and ho is an equivalence of triangulated categories. The proof of Theorem 1.3 is a modification of the argument given for the non-equivariantcase in [LN18]. It will be given in Section 2. The specific motivation for the present paper was the need, in the companion paper [BEL],to refine the classical equivariant locally finite K -homology functor with coefficients in a G - C ∗ -algebra A X (cid:55)→ K G, lf A, ∗ ( X ) := KK G ∗ ( C ( X ) , A )to a spectrum valued functor. Using Theorem 1.3 we get such a refinement by setting X (cid:55)→ K G, lf A ( X ) := KK G sep ( C ( X ) , A ) , (1.3)where for the moment A must be separable, and is X a locally compact and secondcountable topological G -space so that C ( X ) is separable, too. Here KK G sep ( B, A ) is ashort-hand notation for the spectrum map KK G sep (kk G sep ( B ) , kk G sep ( A )). In Definition 1.13below we will remove the restrictions on A and X . Note that the link points to the end of the proof. G sep and the ∞ -category KK G sep whichreflect well-known properties of Kasparov’s bifunctor KK G . We first explain some of thenotions appearing in the statement. We consider a functor from G - C ∗ -algebras to a stable ∞ -category. It is called reduced if it sends the zero algebra to a zero object. It is semiexactif it sends every semisplit exact sequence to a fibre sequence, where an exact sequence issemisplit if it admits an equivariant cpc (completely positive and contractive) split (seealso Definition 2.13.1). It is K G -stable if it sends morphisms of the form A ⊗ K ( H ) → A ⊗ K ( H (cid:48) ) (1.4)to equivalences, where H → H (cid:48) is an equivariant isometric inclusion of non-zero separable G -Hilbert spaces. It is homotopy invariant if it sends the morphisms of the form A → C ([0 , ⊗ A (1.5)given by the embedding a (cid:55)→ a ⊗ A : N → Fun ( BG, C ∗ Alg nusep ). Theorem 1.4. kk G sep is reduced.2. kk G sep is semiexact.3. kk G sep is K G -stable.4. kk G sep is homotopy invariant.5. KK G sep admits countable colimits and is therefore idempotent complete.6. kk G sep preserves countable sums.7. kk G sep preserves colimits of admissible diagrams A : N → Fun ( BG, C ∗ Alg nusep ) . The proof of this theorem will be given in Section 2.The functor (1.1) has a characterization by universal properties for functors to additivecategories [Hig87], [Tho98], [Mey00], see Proposition 2.2. Our next theorem states thatkk G sep has a similar universal property for functors from Fun ( BG, C ∗ Alg nusep ) to objects ofthe large category
Cat ex ∞ of small stable ∞ -categories and exact functors. Theorem 1.5.
The functor kk G sep : Fun ( BG, C ∗ Alg nusep ) → KK G sep is initial among functorsfrom Fun ( BG, C ∗ Alg nusep ) to objects of Cat ex ∞ which are reduced, semiexact and K G -stable. Remark 1.6.
Note that the universal property stated in Theorem 1.5 is different from theobvious one stating that kk G sep is the initial functor to ∞ -categories which inverts KK G sep -equivalences. The latter holds true by definition of KK G sep as a Dwyer–Kan localization.Note also that in most references, the universal property is stated for functors which arein addition homotopy invariant. In the unequivariant situation it was first proven byHigson that homotopy invariance follows from split-exactness and K -stability, and in theequivariant situation the same is true, see also Remark 2.3.In the following, we consider the minimal and maximal tensor products ⊗ min and ⊗ max of C ∗ -algebras. Both of them equip the category Fun ( BG, C ∗ Alg nu ) of possibly non-unital C ∗ -algebras with G -action with a symmetric monoidal structure and preserve separablealgebras. Proposition 1.7 (Proposition 2.20) . The tensor product ⊗ ? for ? in { min , max } descendsto a bi-exact symmetric monoidal structure on KK G sep , and kk G sep refines to a symmetricmonoidal functor kk G, ⊗ ? sep : Fun ( BG, C ∗ Alg nusep ) ⊗ ? → KK G, ⊗ ? sep . ∞ -category for G - C ∗ -algebras For the purpose of the application in [BEL], the restriction of the definition of locally finite K -homology in (1.3) to separable coefficient algebras A is not sufficient. Therefore wemust extend the functor kk G sep from separable C ∗ -algebras to all C ∗ -algebras with G -action.In order to fix size issues we choose an increasing sequence of three Grothendieck universeswhose elements will be called small, large and very large sets. All C ∗ -algebras are assumedto be small. The category Fun ( BG, C ∗ Alg nusep ) is essentially small, and it follows from thedetails of the proof of Theorem 1.3 that KK G sep is also essentially small. In contrast, thecategory Fun ( BG, C ∗ Alg nu ) is large, but locally small. Definition 1.8.
We define the ∞ -category KK G := Ind(KK G sep ) as the Ind -completion of KK G sep and let y G : KK G sep → KK G (1.6) denote the canonical functor. emark 1.9. If C is a small, stable ∞ -category, then by [BGT13, Prop. 3.2] an explicitmodel for the canonical functor C →
Ind( C ) is given by the Yoneda embedding C →
Fun ex ( C op , Sp ) , C (cid:55)→ map C ( − , C ) . The ∞ -category Ind( C ) is compactly generated, presentable and stable. Moreover, thefunctor C →
Ind( C ) ω exhibits the full subcategory Ind( C ) ω of compact objects in Ind( C )as the idempotent completion of C . Definition 1.10.
We define the functor kk G : Fun ( BG, C ∗ Alg nu ) → KK G as the left Kan extension of y G ◦ kk G sep along the inclusion incl as indicated in Fun ( BG, C ∗ Alg nusep ) incl (cid:42) (cid:42) kk G sep (cid:47) (cid:47) KK G sep y G (cid:47) (cid:47) KK G . Fun ( BG, C ∗ Alg nu ) kk G (cid:54) (cid:54) ⇓ (1.7) K -homology Using the Definitions 1.8 and 1.10 we can now extend the definition of the spectrum valuedlocally finite K -homology functor (1.3) to all G - C ∗ -algebras (or even objects of KK G ) A and locally compact G -spaces X .We let G Top prop denote the category of locally compact Hausdorff spaces with G -actionand equivariant, continuous and proper maps. We further consider the category G Top prop+ with the same objects, but with the larger set of maps
Hom G Top prop+ ( X, Y ) :=
Hom G Top prop (( X + , ∞ X ) , ( Y + , ∞ Y )) , where X + and Y + are the one-point compactifications of X and Y , respectively, and a map f : ( X + , ∞ X ) → ( Y + , ∞ Y ) is a continuous equivariant map X + → Y + with f ( ∞ X ) = ∞ Y .Equivalently, a morphism f : X → Y in G Top prop+ is a partially defined map X ⊇ U f → Y on an open subset U of X with f in G Top prop which corresponds to the map f + : X + → Y + such that f + | U = f and f ( X + \ U ) = {∞ Y } .Two morphisms X → Y in G Top prop+ are called properly homotopic if there exists ahomotopy [0 , × X → Y in G Top prop+ between them.The category G Top prop+ is the natural domain of the functor C ( − ) : ( G Top prop+ ) op → Fun ( BG, C ∗ Alg nu )7hich sends a locally compact G -space to the C ∗ -algebra of continuous functions vanishingat ∞ and with the induced G -action. More precisely, C ( X ) is the kernel of the evaluationmap C ( X ) := ker ( C ( X + ) → C ) , where the evaluation map takes the value at the point ∞ . Since we do not assume that X is second countable, the algebra C ( X ) is in general not separable.If Y is an invariant closed subset of X , then we have an exact sequence0 → C ( X \ Y ) → C ( X ) → C ( Y ) → Fun ( BG, C ∗ Alg nu ). Definition 1.11.
Was say that Y is split-closed in X , if the sequence (1.8) is semisplit,i.e. if it admits an equivariant cpc (completely positive and contractive) split. The following proposition provides sufficient criteria for Y being split-closed. Proposition 1.12 (Proposition 5.1) . The closed invariant subset Y of X is split closedin the following cases:1. G acts properly on an invariant neighbourhood of Y in X and Y is second countable.2. Y admits a G -invariant tubular neighbourhood. Definition 1.13.
We define the equivariant, locally finite K -homology functor K G, lf : G Top prop+ × KK G → Sp by ( X, A ) (cid:55)→ K G, lf A ( X ) := KK G ( C ( X ) , A ) . The following theorem states the basic properties of equivariant locally finite K -homology.All spaces in the statement belong to G Top prop+ , and A is in KK G or Fun ( BG, C ∗ Alg nu ),where in the latter case we drop the functor kk G in order to simplify the notation. InRemark 1.15 below we will explain these statements in more detail. Theorem 1.14.
1. If X is second countable and A is a σ -unital G - C ∗ -algebra, then we have an isomor-phism of Z -graded abelian groups K G, lf A, ∗ ( X ) ∼ = KK G ∗ ( C ( X ) , A ) . . The functor K G, lf is homotopy invariant.3. If Y is a split-closed G -invariant subspace of X , then we have a fibre sequence K G, lf A ( Y ) → K G, lf A ( X ) → K G, lf A ( X \ Y ) . (1.9)
4. We consider an exact sequence → A → B → C → in Fun ( BG, C ∗ Alg nu ) . Ifthe sequence is semisplit, or if X is properly homotopy equivalent to a finite G -CWcomplex with finite stabilizers, then we have a fibre sequence K G, lf A ( X ) → K G, lf B ( X ) → K G, lf C ( X ) (1.10) If X is second countable, then KK G (cid:51) A (cid:55)→ K G, lf A ( X ) (1.11) preserves filtered colimits.5. We have K G, lf A ([0 , ∞ ) × X ) (cid:39) . Furthermore, if ( X n ) n ∈ N is a family of second countable spaces and A is separable,then we have a canonical equivalence. K G, lf A (cid:0) (cid:71) n ∈ N X n (cid:1) (cid:39) −→ (cid:89) n ∈ N K G, lf A ( X n ) .
6. If X ⊇ X ⊇ . . . ⊇ X n ⊇ . . . is a decreasing sequence of closed invariant subspacesof a second countable space X such that (cid:84) n X n → X is split-closed, and A isseparable, then we have an equivalence K G, lf A (cid:0) (cid:92) n ∈ N X n (cid:1) (cid:39) −→ lim n ∈ N K G, lf A ( X n ) .
7. If H is a finite subgroup of G , then we have an equivalence K G, lf A ( G/H ) (cid:39) K C ∗ Alg (Res GH ( A ) (cid:111) H ) .
8. The functor K G, lf has a lax symmetric monoidal refinement G Top prop+ , ⊗ × Fun ( BG, C ∗ Alg nu ) ⊗ ? → Sp ⊗ for ? in { min , max } . emark 1.15. In this remark we explain the meaning of the assertions of Theorem 1.14in greater detail.The Assertion 1.14.1 shows that, under the conditions on X and A as stated, the functor K G, lf A is a spectrum valued refinement of the classical equivariant locally finite K -homologyfunctor.The homotopy invariance in X stated in Assertion 1.14.2 first of all means that the functor K G, lf A sends the projection [0 , × X → X to an equivalence. Since we work in the categoryof proper equivariant maps, this implies invariance of K G, lf A under proper equivarianthomotopies. Furthermore, for fixed X the functor K G, lf − ( X ) is also homotopy invariant inthe algebra variable.The Assertion 1.14.3 implies that K G, lf A satisfies excision for invariant split-closed decom-positions ( Z, Y ) of X , i.e., invariant decompositions such that Y is split-closed in X and Y ∩ Z is split-closed in Z . Furthermore, the fact that the fibre of K G, lf A ( Y ) → K G, lf A ( X )only depends on the complement X \ Y is often referred to as the strong excision axiom.Note that the map K G, lf A ( X ) → K G, lf A ( X \ Y ) in (1.9) is induced by the partially definedmap X ⊇ X \ Y id X \ Y −−−→ X \ Y .It immediately follows from Definition 1.13 that for fixed X , the functorKK G (cid:51) A (cid:55)→ K G, lf A ( X ) ∈ Sp preserves limits. The Assertion 1.14.4 states additional exactness properties of K G, lf A ( X )as a functor on Fun ( BG, C ∗ Alg nu ) rather than KK G .The Assertion 1.14.5 is our expression of local finiteness of K G, lf A . The second part is oftenreferred to as the cluster axiom.The Assertion 1.14.6 is also called the continuity axiom.Following [LN18], in Assertion 1.14.7 we use the spectrum valued K -theory functor for C ∗ -algebras K C ∗ Alg ( − ) := KK( C , − ) : C ∗ Alg nu → Sp (1.12)which is equivalent to the one constructed in e.g. [Joa03], [Joa04]; see [LN18, Prop. 3.7.1].Let G Orb be the orbit category of G , i.e. the full subcategory of G -sets consistingof transitive G -sets, and let G Fin
Orb denote its full subcategory on orbits with finitestabilizers. Considering G -sets as discrete topological spaces, we have an embedding G Fin
Orb → G Top prop+ and can define a functor K G, an A : G Fin
Orb → G Top prop+ K G, lf A −−−→ Sp . (1.13)The Assertion 1.14.7 implies that this functor has the same values as the Davis–L¨uckfunctor used in [DL98], [Joa03], [LNS17] (for A = C ) and [Kra] (in general). In [BEL], we10ill upgrade this to an equivalence of functors, see (A.9) for a precise statement. In theAppendix A we explain how the functor K G, an A , which is the domain of the spectrum valuedBaum–Connes assembly map, features in a comparison of assembly maps, see DiagramA.10. Its construction is one of the motivations for the companion paper [BEL].In Assertion 8 the symmetric monoidal structure on G Top prop+ , ⊗ is given by the cartesianproduct of the underlying topological spaces.The functor K G, lf will be derived in (5.6) from the more fundamental functorkk G C := kk G ◦ C : G Top prop+ → KK G whose properties will be stated in Theorem 5.2.The proof of Theorem 1.14 (and of Theorem 5.2) will employ almost all of the generalresults about kk G stated below. It will be completed in Section 5. We now come back to the properties of the functor kk G from Definition 1.10. Definition 1.16.
A functor F defined on Fun ( BG, C ∗ Alg nu ) is called s-finitary if forevery A in Fun ( BG, C ∗ Alg nu ) the canonical map colim A (cid:48) ⊆ A F ( A (cid:48) ) → F ( A ) (1.14) is an equivalence, where A (cid:48) runs through the separable G -invariant subalgebras of A . The prefix ‘s’ stands for separable. In contrast, in the literature a finitary functor is usuallyrequired to preserve all filtered colimits. The following theorem lists the basic propertiesof kk G . Theorem 1.17. kk G is s-finitary.2. kk G is reduced.3. kk G is semiexact.4. kk G is homotopy invariant. . kk G is K G -stable. The proof of this theorem will be finished in Section 3.In order to formulate the universal property of kk G we consider the very large ∞ -categorycategory CAT ccpl ∩ ex ∞ of cocomplete (with respect to small colimits) stable ∞ -categoriesand functors preserving small colimits. As noted in Remark 1.9 the ∞ -category KK G ispresentable stable and therefore belongs to CAT ccpl ∩ ex ∞ . Theorem 1.18 (Theorem 3.3) . kk G is initial among functors from Fun ( BG, C ∗ Alg nu ) to objects of CAT ccpl ∩ ex ∞ which are s-finitary, reduced, K G -stable and semiexact. Assume that
A, B are in
Fun ( BG, C ∗ Alg nu ). Then on the one hand, we have the abeliangroup KK G ( A, B ) defined by Kasparov in [Kas88]. On the other hand, we have the abeliangroup π KK G ( A, B ) defined by the abstract categorical procedure in Definition 1.8. Ifboth A and B are separable, then these two groups are identified by the morphism ho in(2.8). The next proposition extends this isomorphism to all degrees and from separable to σ -unital B . We let C ∗ Alg nu σ be the full subcategory of C ∗ Alg nu of σ -unital C ∗ -algebras. Proposition 1.19 (Proposition 3.5) . For any objects A in Fun ( BG, C ∗ Alg nusep ) and B in Fun ( BG, C ∗ Alg nu σ ) , the functor ho induces an isomorphism of Z -graded abelian groups π ∗ KK G ( A, B ) ∼ = KK G ∗ ( A, B ) . Note that this proposition for separable B is a direct consequence of the compatibility ofho with the triangulated structure stated in Theorem 1.3.2.The following result extends Proposition 1.7 from the separable to the general case. Proposition 1.20 (Proposition 3.8) . The symmetric monoidal structure ⊗ ? on KK G sep for ? in { min , max } canonically induces a presentably symmetric monoidal structure on KK G and kk G refines to a symmetric monoidal functor kk G, ⊗ ? : Fun ( BG, C ∗ Alg nu ) ⊗ ? → KK G, ⊗ ? . As an immediate consequence of Proposition 1.20 we can define for ? in { min , max } aninternal morphism functorkk G ? ( − , − ) : (KK G ) op × KK G → KK G . (1.15)It is characterized by a natural equivalence KK G ( A, kk G ? ( B, C )) (cid:39) KK G ( A ⊗ ? B, C ) forall
A, B, C in KK G and preserves limits in both arguments.12 .5. Change of groups functors In the following we consider various change of groups functors. If H → G is a homomor-phism of groups, then we have an obvious restriction functorRes GH : Fun ( BG, C ∗ Alg nu ) → Fun ( BH, C ∗ Alg nu ) . If H is a subgroup of G , then we have an induction functorInd GH : Fun ( BH, C ∗ Alg nu ) → Fun ( BG, C ∗ Alg nu )which will be explained in Construction 4.6. Finally, we have maximal and reduced crossedproduct functors − (cid:111) max G, − (cid:111) r G : Fun ( BG, C ∗ Alg nu ) → C ∗ Alg nu whose details will be recalled in Construction 4.11. The following results say that thesefunctors descend to functors (denoted by the same symbols) between the correspondingstable ∞ -categories. Theorem 1.21.
1. There exists a factorization
Fun ( BG, C ∗ Alg nu ) Res GH (cid:47) (cid:47) kk G (cid:15) (cid:15) Fun ( BH, C ∗ Alg nu ) kk H (cid:15) (cid:15) KK G Res GH (cid:47) (cid:47) KK H and Res GH : KK G → KK H preserves colimits and compact objects.2. There exists a factorization Fun ( BH, C ∗ Alg nu ) Ind GH (cid:47) (cid:47) kk H (cid:15) (cid:15) Fun ( BG, C ∗ Alg nu ) kk G (cid:15) (cid:15) KK H Ind GH (cid:47) (cid:47) KK G and Ind GH : KK H → KK G preserves colimits and compact objects.3. There exists a factorization Fun ( BG, C ∗ Alg nu ) (cid:111) ? G (cid:47) (cid:47) kk G (cid:15) (cid:15) C ∗ Alg nukk (cid:15) (cid:15) KK G − (cid:111) ? G (cid:47) (cid:47) KK for ? ∈ { r, max } and − (cid:111) ? G : KK G → KK preserves colimits and compact objects. G sep is idempotent complete by Theorem 1.4.5, the functor y G in (1.6) identifiesKK G sep with the full subcategory of KK G of compact objects, see Remark 1.9. Hence theassertion that the functors preserve compact objects means that they induce functorsbetween the separable versions of the respective KK-categories. It then follows from theassertion about preservation of colimits that the functors appearing in Theorem 1.21 areequivalent to the canonical extensions of their separable versions. The proof of Theorem1.21 will be given in Section 4.There exists a canonical natural transformation ι : id → Res GH ◦ Ind GH , (1.16)explained in detail in (4.3), of endofunctors of Fun ( BH, C ∗ Alg nu ). It looks like the unitof an adjunction, and it indeed becomes the unit of the adjunction after application ofkk G (and it is not one before), see (1.20) below. The transformation ι induces the firsttransformation of functors in( − ) (cid:111) ? H → Res GH ◦ Ind GH ( − ) (cid:111) ? H → Ind GH ( − ) (cid:111) ? G , (1.17)where the second is canonically induced by the inclusion of H into G , see (4.11).We now assume that H is a finite group. If A is in C ∗ Alg nu , then we consider thehomomorphism (cid:15) A : A → Res H ( A ) (cid:111) H , a (cid:55)→ | H | (cid:88) h ∈ H ( a, h ) , where Res H ( A ) denotes A equipped with the trivial H -action. The family (cid:15) = ( (cid:15) A ) A ∈ C ∗ Alg nu is a natural transformation (cid:15) : id → Res H ( − ) (cid:111) H (1.18)of endofunctors of C ∗ Alg nu .Let B be in C ∗ Alg nu . Then we have a canonical homomorphism λ B : Res G ( B ) (cid:111) max G → B of C ∗ -algebras which corresponds to the covariant representation ( id B , triv) consisting ofthe identity of B and the trivial representation of G . The family λ = ( λ B ) B ∈ C ∗ Alg nu is anatural transformation λ : Res G ( − ) (cid:111) max G → id (1.19)of endofunctors of C ∗ Alg nu .The Assertions 1 and 3 of the following theorem are ∞ -categorical level versions of Green’sImprimitivity Theorem [Gre78] (for ? = max), [Kas88] and the Green–Julg theorem [Jul81].Both statements generalize the versions for the triangulated categories stated in [MN06].Assertion 4 is known as the dual Green–Julg theorem.14 heorem 1.22.
1. The natural transformation (1.16) induces the unit of an adjunction
Ind GH : KK H (cid:29) KK G : Res GH . (1.20)
2. The transformation (1.17) naturally induces an equivalence of functors ( − ) (cid:111) ? H → Ind GH ( − ) (cid:111) ? G : KK H → KK for ? in { r, max } .3. If H is finite, then the natural transformation (1.18) induces the unit of an adjunction Res H ( − ) : KK (cid:29) KK H : − (cid:111) H .
4. The natural transformation (1.19) induces the counit of an adjunction − (cid:111) max G : KK G (cid:29) KK : Res G ( − ) . Since H is finite, the crossed product in Assertion 3 need not be decorated. It is simplythe algebraic crossed product which happens to be equal to the reduced and maximal onein this case. Note that Assertion 2 in the case ? = max is also a consequence of Assertions1 and 4. The proof of this theorem will be completed in Section 4.Note that Ind GH ( C ) ∼ = C ( G/H ). Let r GH : KK G ( C ( G/H ) , − ) (cid:39) KK H ( C , Res GH ( − )) (1.21)denote the equivalence of the mapping spectrum functors given by the adjunction in theTheorem 1.22.1. Furthermore, letGJ H : KK H ( C , Res GH ( − )) (cid:39) KK( C , Res GH ( − ) (cid:111) H )denote the equivalence of the mapping spectrum functors given by the adjunction in theTheorem 1.22.3. Corollary 1.23. If H is a finite subgroup of G , then we have an equivalence GJ H ◦ r GH : KK G ( C ( G/H ) , − ) → K C ∗ Alg (Res GH ( − ) (cid:111) H ) (1.22) of functors from Fun ( BG, C ∗ Alg nu ) to Sp .
15y Theorem 1.17.3 the functor kk G sends exact sequences in Fun ( BG, C ∗ Alg nu ) admittingequivariant cpc splits to fibre sequences. As an immediate consequence, the functorKK G ( A, − ) : Fun ( BG, C ∗ Alg nu ) → Sp has the same property for every A in KK G . The Theorem 1.26 shows that under certainrestrictions on A this functor in fact sends all exact sequences to fibre sequences. Theconditions are formulated so that we can deduce this exactness using Corollary 1.23 fromthe fact that the usual K -theory functor for C ∗ -algebras in (1.12) sends all exact sequencesto fibre sequences.Recall that a thick subcategory of a stable ∞ -category is a full stable subcategory which isclosed under taking retracts. A localizing subcategory of a presentable stable ∞ -categoryis a full stable subcategory which is closed under all colimits. Definition 1.24.
1. The objects of the thick subcategory of KK G generated by the objects kk G ( C ( G/H )) for all finite subgroups H of G are called G -proper.2. The objects of the localizing subcategory of KK G generated by the G -proper objectsare called ind- G -proper. Note that G -proper objects are ind- G -proper. The following result provides examples of G -proper objects in KK G . Proposition 1.25 (Proposition 1.19) . If X is in G Top prop and homotopy equivalent (in G Top prop ) to a finite G -CW complex with (necessarily) finite stabilizers, then kk G ( C ( X )) is G -proper. Let P be an object of kk G . Theorem 1.26 (Theorem 5.6) .
1. If P is ind- G -proper, then the functor KK G ( P, − ) : Fun ( BG, C ∗ Alg nu ) → Sp sends all exact sequences to fibre sequences.2. If P is G -proper, then the functor KK G ( P, − ) : Fun ( BG, C ∗ Alg nu ) → Sp preserves filtered colimits. .6. Extensions to C ∗ -categories Again motivated by the applications in [BEL], we extend the functor kk G from C ∗ -algebrasto C ∗ -categories with G -action. We refer to the beginning of Section 6 for a more detailedintroduction to C ∗ -categories. We have a fully faithful inclusion of the category C ∗ Alg nu of(possibly non-unital) C ∗ -algebras into the category C ∗ Cat nu of small (possibly non-unital) C ∗ -categories which considers a C ∗ -algebra as a C ∗ -category with a single object. Thisinclusion fits into an adjunction A f : C ∗ Cat nu (cid:29) C ∗ Alg nu : incl (1.23)first considered by [Joa03] in the unital case, and extended to the non-unital case in[Bun]. Definition 1.27.
We define the functor kk GC ∗ Cat : Fun ( BG, C ∗ Cat nu ) → KK G as the composition Fun ( BG, C ∗ Cat nu ) A f −→ Fun ( BG, C ∗ Alg nu ) kk G −−→ KK G . The following extends Definition 1.16 from C ∗ -algebras to C ∗ -categories. Definition 1.28.
A functor F defined on Fun ( BG, C ∗ Cat nu ) is called s-finitary, if forevery C in Fun ( BG, C ∗ Cat nu ) the canonical map colim C (cid:48) F ( C (cid:48) ) → F ( C ) (1.24) is an equivalence, where C (cid:48) runs through the G -invariant subcategories of C which haveat most countably many non-zero objects and separable morphism spaces. As in the case of C ∗ -algebras, we added the letter ‘s’ (for separable), since this notiondiffers from the definition of a finitary functor in [BEa, Def. 11.16]. The latter requiresthe preservation of all filtered colimits.In the following theorem we list the properties of the functor kk GC ∗ Cat for C ∗ -categories. Theorem 1.29.
1. The functor kk GC ∗ Cat is s-finitary.2. The functor kk GC ∗ Cat sends unitary equivalences in
Fun ( BG, C ∗ Cat ) to equivalences. . The functor kk GC ∗ Cat sends weak Morita equivalences to equivalences.4. We have an equivalence kk C ∗ Cat ( − (cid:111) ? G ) (cid:39) ( − (cid:111) ? G ) ◦ kk GC ∗ Cat of functors
Fun ( BG, C ∗ Cat nu ) → KK for ? ∈ { r, max } .5. If P in KK G is ind- G -proper, then the functor KK G ( P, kk GC ∗ Cat ( − )) sends all exactsequences in Fun ( BG, C ∗ Cat nu ) to fibre sequences.6. If P in kk G is G -proper, then KK G ( P, kk GC ∗ Cat ( − )) preserves filtered colimits.7. If C in Fun ( BG, C ∗ Cat ) is flasque, then KK G ( P, kk GC ∗ Cat ( C )) (cid:39) for all ind- G -proper P in KK G . The proof of Theorem 1.29 will be given in Section 6.
Remark 1.30.
Let us consider here the case of the trivial group. In [BEa, Def. 11.5] weintroduced the notion of a homological functor Hg : C ∗ Cat nu → M . One can characterizea homological functor by the conditions that M is stable, and that Hg sends unitaryequivalences between unital C ∗ -categories to equivalences, is reduced (sends zero categoriesto zero), and sends exact sequences of C ∗ -categories to fibre sequences. If Hg preservesfiltered colimits, then it is called finitary.If P in KK is ind-proper, then it follows from the Theorem 1.29 and the obvious fact thatkk C ∗ Cat is reduced that KK( P, kk C ∗ Cat ( − )) : C ∗ Cat nu → Sp is a homological functor. If P is proper, then this functor is also finitary.As a example, if X is locally compact and homotopy equivalent to a finite CW-complex,then P := kk( C ( X )) is proper by Proposition 1.25, and therefore KK( C ( X ) , kk C ∗ Cat ( − ))is a homological functor.The functor kk C ∗ Cat itself is not a KK-valued homological functor since it does not sendall exact sequences to fibre sequences.The minimal and maximal tensor products for C ∗ -algebras can naturally be extendedto C ∗ -categories, see [Del12] for ⊗ max , and [AV] for both. In Section 7 we will give acomprehensive treatment of both cases. The definition of the maximal tensor product interms of its universal property is stated in Definition 7.2, while the analoguous definitionof the minimal tensor product is given in Definition 7.5. Theorem 1.31.
The functor kk GC ∗ Cat refines to a symmetric monoidal functor kk G, ⊗ ? C ∗ Cat : Fun ( BG, C ∗ Cat nu ) ⊗ ? → KK G, ⊗ ? or ? in { min , max } . The proof of this theorem will be given in Section 7.
2. A stable ∞ -category of separable G - C ∗ -algebras Let C ∗ Alg nusep and C ∗ Alg nu σ denote the full subcategories of C ∗ Alg nu of separable andof σ -unital C ∗ -algebras, respectively. In the present paper we work with the bivariant G -equivariant Kasparov KK -theory functor [Kas88] KK G ∗ : Fun ( BG, C ∗ Alg nu σ ) op × Fun ( BG, C ∗ Alg nu σ ) → Ab Z , where Ab Z denotes the category of Z -graded abelian groups. Further good references forthis functor and its properties are [Mey00] and [Bla98].We will use the symbol KK G := KK G for the Ab -valued functor obtained from KK G ∗ byextracting the degree-zero component. Using the Kasparov intersection product in orderto define the composition we construct the Ab -enriched category KK G and the functorkk G : Fun ( BG, C ∗ Alg nusep ) → KK G (appearing in (1.1)) as explained in the introduction. The functor kk G is reduced, homotopyinvariant and K G -stable since it inherits these properties from KK G .The functor kk G can be characterized by the following universal property: Proposition 2.1.
The functor kk G : Fun ( BG, C ∗ Alg nusep ) → KK G exhibits the target aslocalisation of Fun ( BG, C ∗ Alg nusep ) at the set of KK G -equivalences in the sense of ordinarycategories.Proof. Let F : Fun ( BG, C ∗ Alg nusep ) → D be any functor to an ordinary category D which sends KK G -equivalences to isomorphismsin D . We must show that there exists a unique factorisation through KK G , indicated bythe dashed arrow in the following diagram Fun ( BG, C ∗ Alg nusep ) F (cid:47) (cid:47) kk G (cid:15) (cid:15) D . KK G F (cid:55) (cid:55) (2.1)19ince kk G : Fun ( BG, C ∗ Alg nusep ) → KK G is the identity on objects we are forced to define¯ F ( A ) := F ( A ) for all objects A of Fun ( BG, C ∗ Alg nusep ).For the discussion of morphisms we use the Cuntz picture of KK G due to [Mey00, Sec. 6].Let K := K ( (cid:96) ⊗ L ( G )) (2.2)denote the algebra of compact operators on the G -Hilbert space (cid:96) ⊗ L ( G ), where G actson L ( G ) by the left-regular representation, and by conjugation on the operators. Wefurthermore let q s ( A ) := ker ( A ⊗ K (cid:116) A ⊗ K → A ⊗ K )be the kernel of the fold map. Note that the coproduct in C ∗ -algebras is realized by thefree product. By [Mey00, Thm. 6.5] we have a bijection KK G ( A, B ) ∼ = [ q s ( A ) ⊗ K, q s ( B ) ⊗ K ] , (2.3)where the notation [ − , − ] stands for norm-continuous homotopy classes of morphisms in Fun ( BG, C ∗ Alg nusep ). Moreover, the composition in KK G corresponds to the compositionof homotopy classes.We consider the separable G -Hilbert space H (cid:48) := C ⊕ (cid:96) ⊗ L ( G ) and the associated G - C ∗ -algebra K (cid:48) := K ( H (cid:48) ) of compact operators. The equivariant inclusion C → H (cid:48) induces an inclusion d : C → K (cid:48) in Fun ( BG, C ∗ Alg nusep ), where C has the trivial G -action.We also have a canonical inclusion e : K → K (cid:48) .Let π A : q s ( A ) → A ⊗ K be the restriction of id A ⊗ K (cid:116) q s ( A ). It is shown in [Mey00,Sec. 6] that π A is a KK G -equivalence. For every A in Fun ( BG, C ∗ Alg nusep ) we have azig-zag q s ( A ) ⊗ K π A ⊗ id K → A ⊗ K ⊗ K id A ⊗ e ⊗ id K → A ⊗ K (cid:48) ⊗ K id A ⊗ d ⊗ id K ← A ⊗ K id A ⊗ e → A ⊗ K (cid:48) id A ⊗ d ⊗ id K ← A (2.4)which is natural in A . By the K G -stability of kk G all these maps are KK G -equivalences.20e now consider the diagram (2.5) Hom D ( A, B ) (2.4) ∼ = Hom
Fun ( BG,C ∗ Alg nusep ) ( A, B ) F (cid:111) (cid:111) kk G (cid:47) (cid:47) KK G ( A, B ) ¯ F A,B (cid:117) (cid:117) (2.4) ∼ = Hom D ( q s ( A ) ⊗ K, q s ( B ) ⊗ K ) Hom
Fun ( BG,C ∗ Alg nusep ) ( q s ( A ) ⊗ K, q s ( B ) ⊗ K ) !!! F (cid:111) (cid:111) !kk G (cid:47) (cid:47) (cid:15) (cid:15) KK G ( q s ( A ) ⊗ K, q s ( B ) ⊗ K ) !!¯ F (cid:99) (cid:99) [ q s ( A ) ⊗ K, q s ( B ) ⊗ K ] F (cid:48) (cid:108) (cid:108) (2.3) ∼ = (cid:50) (cid:50) The left (or right) vertical isomorphism is induced by applying F (or kk G ) to the zig-zag(2.4) using the fact that F (or kk G ) sends kk G -equivalences to isomorphisms. If ¯ F exists,then the outer square and the two triangles involving dashed and bold arrows commute.Using the lower right triangle we see that the arrow marked by ! is surjective. This showsthat the arrow ¯ F marked by !! is uniquely determined if it exists.Since F inverts KK G -equivalences, it also sends homotopic maps to equal maps. Conse-quently, the arrow F marked by !!! factorizes uniquely through the dotted arrow F (cid:48) asindicated. We are therefore forced to define¯ F A,B : KK G ( A, B ) → Hom D ( A, B )as the composition going clockwise from the upper right-corner to the upper left corner.The whole diagram is bi-natural in A (contravariant) and B (covariant). This impliesthat the family ( ¯ F A,B ) A,B ∈ Fun ( BG,C ∗ Alg nusep ) provides the action of the desired functor ¯ F onmorphisms. The triangle (2.1) then commutes by construction.The following universal property of kk G (which differes from the one in Proposition2.1) was shown in [Tho98, Thm. 2.2], see also [Mey00, Thm. 6.6]. A functor from Fun ( BG, C ∗ Alg nusep ) to an additive category is called split exact if it preserves split-exactsequences, see also Definition 2.13.2.
Proposition 2.2.
The functor kk G : Fun ( BG, C ∗ Alg nusep ) −→ KK G is initial among all functors to an additive -category which are reduced, K G -stable andsplit exact. emark 2.3. The statement of Meyer requires additivity and homotopy invariance of thefunctors. But note that the condition of being split exact and additive are equivalent tothe condition of being split exact and reduced. Furthermore, we use that split-exactnessand K G -stability together imply homotopy invariance (see [Hig88] for the non-equivariantcase). We refer to [CMR07, Thm. 3.35] for an argument which applies verbatim to theequivariant case. Corollary 2.4.
A functor from
Fun ( BG, C ∗ Alg nusep ) to an additive ∞ -category which isreduced,homotopy invariance deleted K G -stable and split exact sends KK G -equivalences toequivalences.Proof. Let F : Fun ( BG, C ∗ Alg nusep ) → D be a functor as in the statement of the corollary.Since ho( D ) is an additive 1-category we can apply Proposition 2.2 to the composition Fun ( BG, C ∗ Alg nusep ) F → D → ho( D ) in order to conclude that it factorizes over kk G . Inparticular, it sends KK G -equivalences to isomorphisms. Since the canonical functor D → ho( D ) detects equivalences we conclude that F sends KK G -equivalences to equivalences. Remark 2.5.
Let C be a small ∞ -category and W be a set of morphisms in C . Then wecan form the Dwyer–Kan localization [Lur, Def. 1.3.4.1, Rem. 1.3.4.2] (cid:96) : C → C [ W − ] . (2.6)It is characterized by the universal property that for any ∞ -category D the restrictionalong (cid:96) induces an equivalence Fun ( C [ W − ] , D ) (cid:39) −→ Fun W ( C , D ) , (2.7)where Fun W ( C , D ) is the full subcategory of Fun ( C , D ) of functors which send morphismsin W to equivalences.Note that we consider ordinary categories as ∞ -categories using the nerve, but we will notwrite the nerve explicitly. With these conventions we can consider the functorkk G sep : Fun ( BG, C ∗ Alg nusep ) → KK G sep introduced in Definition 1.2.We now start with the proof of Theorem 1.3 following the lines of [LN18] using [Mey00] and[Uuy13]. First of all it follows from the universal property of the Dwyer–Kan localizationthat there exists a factorization of kk G over a functor ho as indicated by the followingcommuting triangle: Fun ( BG, C ∗ Alg nusep ) kk G (cid:47) (cid:47) kk G sep (cid:40) (cid:40) KK G . KK G sep ho (cid:59) (cid:59) (2.8)22his is the commutative triangle in (1.2).In order to show that kk G sep is stable we need the explicit model for the suspension functor S : Fun ( BG, C ∗ Alg nusep ) → Fun ( BG, C ∗ Alg nusep ) , A (cid:55)→ C ((0 , ⊗ A . (2.9)
Lemma 2.6.
1. The functor S uniquely descends to an equivalence S of additive categories such that Fun ( BG, C ∗ Alg nusep ) S (cid:47) (cid:47) kk G (cid:15) (cid:15) Fun ( BG, C ∗ Alg nusep ) kk G (cid:15) (cid:15) KK G S (cid:47) (cid:47) KK G (2.10) commutes.2. The functor S essentially uniquely descends to a functor (also denoted by) S com-pleting the square Fun ( BG, C ∗ Alg nusep ) kk G sep (cid:15) (cid:15) S (cid:47) (cid:47) Fun ( BG, C ∗ Alg nusepkk G sep (cid:15) (cid:15) KK G sep S (cid:47) (cid:47) KK G sep (2.11) Proof.
In order to obtain S , we apply Proposition 2.2 to the compositionkk G ◦ S : Fun ( BG, C ∗ Alg nusep ) → KK G . It straightforward to check that the functor kk G ◦ S is reduced, K G -stable and split exact,using that kk G has these properties. The functor kk G ◦ S therefore factorizes over anadditive endofunctor S as required. It then follows from Bott periodicity that S is anequivalence of categories.Since by definition kk G detects KK G -equivalences, we conclude from Assertion 1 that S in (2.9) preserves kk G -equivalences. Applying the universal property of kk G sep being aDwyer-Kan localization (see Definition 1.2) we obtain the essentially unique factoriaztionof kk G sep ◦ S through kk G sep as asserted in Assertion 2.As shown in [Kas88], for every semisplit exact sequence0 → I → A → B → ∂ in KK G ( S ( B ) , I ) such that the sequence KK G ( D, S ( A )) → KK G ( D, S ( B )) ∂ → KK G ( D, I ) → KK G ( D, A ) → KK G ( D, B ) (2.13)23s exact for every D in Fun ( BG, C ∗ Alg nusep ). The boundary operator is natural in thesequence in the following sense. If0 (cid:47) (cid:47) I (cid:15) (cid:15) (cid:47) (cid:47) A (cid:47) (cid:47) (cid:15) (cid:15) B (cid:47) (cid:47) (cid:15) (cid:15) (cid:47) (cid:47) I (cid:48) (cid:47) (cid:47) A (cid:48) (cid:47) (cid:47) B (cid:48) (cid:47) (cid:47) KK G ( D, S ( B )) ∂ (cid:47) (cid:47) (cid:15) (cid:15) KK G ( D, I ) (cid:15) (cid:15) KK G ( D, S ( B (cid:48) )) ∂ (cid:48) (cid:47) (cid:47) KK G ( D, I (cid:48) )commutes.The main problem in the proof of Theorem 1.3 is to control finite limits in KK G sep . To thisend we calculate the Dwyer-Kan localization in Definition 1.2 using a category of fibrantobjects structure on Fun ( BG, C ∗ Alg nusep ). We now recall the definition of a category offibrant objects in the form used in [Uuy13, Def. 1.1]. Let C be a category with a terminalobject and F and W be collections of morphisms in C . Definition 2.7.
The triple ( C , F, W ) is a category of fibrant objects if the followingconditions hold.1. F0: F is closed under compositions.2. F1: F contains all isomorphisms.3. F2: Pull-backs of morphisms in F exist and belong again to F .4. F3: For every C in C the morphism C → ∗ belongs to F .5. W1: W contains all isomorphisms.6. W2: W has the -out-of- -property.7. FW1: W ∩ F is stable under forming pull-backs.8. FW2: For every C in C the morphism C → C × C has a factorization C w → C I f → C × C , where w is in W and f is in F (such data are called a path object). Note that the product in the formulation of Condition FW2 exists by a combination ofConditions F3 and F2. 24 emark 2.8.
The object C I appearing in FW2 is called a path object. From the existenceof a path object C I one can construct a factorisation of an arbitrary morphism φ : A → C in C as a weak equivalence followed by a fibration as follows. We choose a factorization C w → C I f → C × C as in F W
2. Then we consider the commuting diagram
A A × C C I C I A × C C × CC u φ f (cid:48) f pr φ × id (2.14)where the square is a pull-back. It exists by F f (cid:48) is a fibration.Using F F pr is also a fibration. We conclude by F pr ◦ f (cid:48) is a fibration. Furthermore, the map u is given by A ∼ = A × C C A × w → A × C C I and therefore a weak equivalence by F W F A → ∗ ). Consequently A u −→ A × C C I pr ◦ f (cid:48) −→ C (2.15)is the desired factorization of φ as a composition of a weak equivalence and a fibration. Remark 2.9.
The category
Fun ( BG, C ∗ Alg nusep ) admits a canonical path object C (cid:55)→ C ([0 , , C ). The factorization C −→ C ([0 , , C ) −→ C × C required in F W C I → C × C is surjectiveand admits an equivariant contractive completely positive split. For example, one cantake the split ( c , c ) (cid:55)→ ( t (cid:55)→ (1 − t ) c + tc ) . (2.16)Applying the general factorization (2.15) to a morphism A → C , one obtains the usualcone sequence 0 −→ C ( φ ) −→ Cyl( φ ) −→ C → . Explicitly, Cyl( φ ) := { ( a, c ) ∈ A ⊕ C ([0 , , C ) | c (0) = φ ( a ) } , (2.17)the homomorphism Cyl( φ ) → C is given by ( a, c ) (cid:55)→ c (1), and C ( φ ) := { ( a, c ) ∈ Cyl( φ ) | c (1) = 0 } with the induced G -actions. 25 morphism φ : A → C in Fun ( BG, C ∗ Alg nusep ) is called a semisplit surjection if it is asurjection and admits an equivariant cpc split. We let SS denote the set of semisplitsurjections. Furthermore recall that W kk G is the set of KK G -equivalences. Proposition 2.10.
The triple ( Fun ( BG, C ∗ Alg nusep ) , SS , W kk G ) is a category of fibrantobjects.Proof. The axioms F F F W
1, and W F
2. We consider a pullback A (cid:47) (cid:47) ψ (cid:15) (cid:15) C φ (cid:15) (cid:15) B (cid:47) (cid:47) D (2.18)in Fun ( BG, C ∗ Alg nusep ) such that φ is in SS with equivariant cpc split s : D → C . Thesquare of the underlying Banach spaces is a pullback square of Banach spaces. Thus theequivariant contractive split s of φ induces an equivariant contractive split t : B → A . Wenow use the fact that an element in A is positive if its images in C and B are positive. Thesame is true for the extension of the square to finite matrices. Using that s is completelypositive we can now conclude that t is completely positive as well. Hence ψ is in SS, too.To show F W
2, we make use of the cylinder discussed in Remark 2.9. The map C → C I is ahomotopy equivalence, hence in particular a KK G -equivalence. Furthermore, as explainedin Remark 2.9 the projection C I → C × C has an equivariant contractive completelypositive split and therefore belongs to SS.In order to prepare the proof of F W −→ I −→ A ψ −→ B −→ . (2.19)As explained above, then we have the boundary operator ∂ in KK G ( S ( B ) , I ) such thatthe sequence KK G ( D, S ( A )) → KK G ( D, S ( B )) ∂ → KK G ( D, I ) → KK G ( D, A ) ψ ∗ → KK G ( D, B ) (2.20)is exact for every D in Fun ( BG, C ∗ Alg nusep ). By the Yoneda Lemma the morphism ψ is aKK G -equivalence if and only if ψ ∗ is an isomorphism for every D in Fun ( BG, C ∗ Alg nusep ).We conclude from the exactness of (2.20) that this is the case if and only if KK G ( D, I ) = 0for every D .We now consider a pull-back square (2.18) in Fun ( BG, C ∗ Alg nusep ) such that φ belongsto SS ∩ W kk G . By F ψ is in SS and it remains to show that ψ belongsto W kk G . This follows from the fact that ker ( φ ) ∼ = ker ( ψ ). Since φ is in W kk G we have KK G ( D, ker ( ψ )) = KK G ( D, ker ( φ )) = 0 for all D in Fun ( BG, C ∗ Alg nusep ). Consequently, ψ belongs to W kk G . 26or the formulation of universal properties it turned out to be useful to reformulate variousexactness properties of functors defined on G - C ∗ -algebras using squares so that they makesense not only for stable targets, but also for pointed ones. But we will show in Lemma2.14 below that these new definitions are equivalent to the old same-named conditions inthe situations where they have been used above. We do these considerations at this pointsince we will use squares already in the proof of Proposition 2.15 below.We consider a cartesian square A (cid:47) (cid:47) p (cid:15) (cid:15) B q (cid:15) (cid:15) C (cid:47) (cid:47) D (2.21)in Fun ( BG, C ∗ Alg nu ). We will consider the following two conditions on the square in(2.21). Definition 2.11.
1. The square (2.21) is called semisplit if there exist equivariant cpc’s s : C → A and t : D → B such that p ◦ s = id C and q ◦ t = id D .2. The square (2.21) is called split if there exist equivariant morphisms s : C → A and t : D → B such that p ◦ s = id C and q ◦ t = id D . Remark 2.12.
The conditions on q appearing in Definition 2.11 imply the correspondingconditions on p , see the verification of the axiom F2 in the proof of Proposition 2.10 fora proof of this fact. In Definition 2.11 the latter could therefore be omitted in all threecases.Note that an exact sequence 0 → I → A q −→ B → Fun ( BG, C ∗ Alg nu ) is semisplit or split, if and only if the cartesian square I (cid:47) (cid:47) (cid:15) (cid:15) A q (cid:15) (cid:15) (cid:47) (cid:47) ˜ B (2.23)has the corresponding property as defined in Definition 2.11.Let F : Fun ( BG, C ∗ Alg nusep ) → D (or F : Fun ( BG, C ∗ Alg nu ) → D ) be a functor to apointed ∞ -category D . Corresponding to the list of conditions in Definition 2.11 weintroduce the following two exactness conditions on the functor F . Definition 2.13. . F is semiexact if it is reduced and sends any semisplit cartesian square to a cartesiansquare in D .2. F is split exact if it is reduced and sends any split cartesian square to a cartesiansquare in D . Let F be a functor as in Definition 2.13. Lemma 2.14.
1. If D is stable, then F is semiexact in the sense of Definition 2.13.1 if and only if F sends all semisplit exact sequences to fibre sequences.2. If D is additive, then F is split exact in the sense of Definition 2.13.2 if and only if F sends all split exact sequences to fibre sequences.Proof. In each case, the ”only if“ implication is obvious. To see the ”if“ implication, weextend the square (2.21) to the left by adding a pullback of p along 0 → C : I A B C D p q
The left and the outer squares are exact sequences which are send by F to fibre sequences.Thus applying F we obtain the diagram F ( I ) F ( A ) F ( B )0 F ( C ) F ( D ) F ( p ) F ( q ) (2.24)in which the left and the outer squares are cartesian and the lower left corner is a zeroobject. In particular we can conclude that F is reduced. We must deduce that the rightsquare is cartesian, too.If D is stable, then we know that left and outer squares are also cocartesian, and hencethe right square is cocartesian as well. Using again that D is stable, we deduce that theright square is cartesian as needed.In the case that D is additive, in order to argue as above, we need additional assumptionsensuring that certain cartesian squares are also cocartesian and vice versa. In fact will usethat a square X YZ W p q
28n an additive ∞ -category where f and g admit sections and fibres, is cartesian if andonly if it is cocartesian. This can be applied to the corresponding squares in (2.24). Infact, F ( p ) and F ( q ) have splits by our assumptions on the right square in (2.24), and theyhave fibres since the left and the outer square in (2.24) are cartesian as noted above.The following proposition shows all assertions of Theorem 1.3 except the one abouttriangulated structures which will be obtained later in Proposition 2.18. Proposition 2.15.
1. The functor kk G sep is semiexact.2. The functor ho : ho(KK G sep ) → KK G is an equivalence.3. The ∞ -category KK G sep is stable.Proof. Assertion 1 is an immediate consequence of the fact that KK G sep is the ∞ -categoryassociated to a category of fibrant objects whose fibrations are the semisplit exact surjec-tions. Indeed, consider a semisplit cartesian square (2.21). Since p and q are fibrationsthis square represents a cartesian square in the ∞ -category KK G sep , see [Cis19, Prop. 7.5.6].We now prove Assertion 2. We will use that forming localizations is compatible with goingover to homotopy categories. Let C be an ∞ -category with a set of morphisms W . Thenhave a commutative square C (cid:96) (cid:47) (cid:47) (cid:15) (cid:15) C [ W − ] (cid:15) (cid:15) ho( C ) ho( (cid:96) ) (cid:47) (cid:47) ho( C [ W − ]) (2.25)where ho( (cid:96) ) : ho( C ) → ho( C [ W − ]) presents its target as the localization of ho( C ) at theset ho( W ) in the sense of ordinary categories. We apply this to Fun ( BG, C ∗ Alg nusep ) inplace of C . Since Fun ( BG, C ∗ Alg nusep ) is an ordinary category the left vertical functor in(2.25) is an equivalence. Hence the functor
Fun ( BG, C ∗ Alg nusep ) kk G sep −−−→ KK G sep → ho(KK G sep )presents its target as the localization of Fun ( BG, C ∗ Alg nusep ) at the set of KK G -equivalencesin the sense of ordinary categories. By Proposition 2.1 the functorkk G : Fun ( BG, C ∗ Alg nusep ) → KK G has the same universal property. This implies that ho : ho(KK G sep ) → KK G is an equiva-lence. 29he proof Assertion 3 that KK G sep is stable can now be copied from [LN18, Prop. 3.3]. Firstwe note that KK G sep , being the ∞ -category associated to a category of fibrant objects witha zero object, is pointed and has finite limits. It remains to show that the loop functorΩ in KK G sep is an equivalence. It is well-known that it suffices to show that Ω induces anequivalence on the homotopy category, see for instance [LN18, Lem. 3.4].Using the explicit description of the fibrant replacement of 0 → A given in Remark 2.9 wesee that kk G sep ( S ( A )) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) kk G sep ( A ) (2.26)is a pull-back in KK G sep . Therefore Ω is equivalent to the suspension functor given by thedotted arrow S in (2.11). In view of Assertion 2 shown above the induced action of thisfunctor S on the homotopy category can be identified with the action of the functor S inLemma 2.6.2 on KK G . As asserted in the same statement S is an equivalence.Our next task is the verification of the properties of KK G sep and kk G sep stated in Theorem1.4. Assertion 1.4.2 is already shown as a part of Proposition 2.15.The next Lemma settles Assertions 1.4.1, 1.4.4 and 1.4.5 together. Lemma 2.16.
The functor kk G sep is reduced, K G -stable, and homotopy invariant.Proof. We know that kk G is reduced, K G -stable and homotopy invariant. ThereforeProposition 2.15.2 together with the fact that the canonical functor from an ∞ -categoryto its homotopy category detects equivalences and zero elements implies the assertion.The next lemma collects, for later reference, some simple consequences of the details proofof Proposition 2.15. Lemma 2.17.
1. Every morphism in KK G sep is equivalent to a morphism kk G sep ( f ) for a fibration f in Fun ( BG, C ∗ Alg nusep ) .2. Every product in KK G sep is equivalent to the image under kk G sep of a product in Fun ( BG, C ∗ Alg nusep ) .3. Every fibre sequence in KK G sep is equivalent to the image under kk G sep of a conesequence. roof. These assertions follow from general facts about the associated ∞ -category of acategory of fibrant objects, where for 3 we use in addition that KK G sep is stable. Forcompleteness we give alternative arguments which are specific to the present situation aswe will use some of the details later.Assertion 1 follows from the equivalence ho : ho(KK G sep ) (cid:39) → KK G and the surjectivity ofthe arrow marked by !! in (2.5).For Assertion 2 we note that any object in KK G sep is equivalent to an object of the formkk G sep ( A ) for some A in Fun ( BG, C ∗ Alg nusep ). Let kk G sep ( B ) be a second object. Then thesquare A ⊕ B (cid:47) (cid:47) (cid:15) (cid:15) A (cid:15) (cid:15) B (cid:47) (cid:47) Fun ( BG, C ∗ Alg nusep ) (Definition 2.11.2). Since kk G sep is semiexactby Prop. 2.151 (and hence split exact) we conclude that the squarekk G sep ( A ⊕ B ) (cid:47) (cid:47) (cid:15) (cid:15) kk G sep ( A ) (cid:15) (cid:15) kk G sep ( B ) (cid:47) (cid:47) G sep ( A ) × kk G sep ( B ) (cid:39) kk G sep ( A ⊕ B ) . (2.28)The assertion now follows fromkk G sep ( A × B ) (cid:39) kk G sep ( A ⊕ B ) (2.28) (cid:39) kk G sep ( A ) × kk G sep ( B ) , where we use that the product of C ∗ -algebras is given by the sum.We finally show Assertion 3. Since KK G sep is stable, a morphism in KK G sep can be extendedto a fibre sequence, and every fibre sequence is obtained in this way up to equivalence.By Assertion 1 it suffices to consider fibre sequences obtained by extending the morphismkk G sep ( p ) for a morphism p : A → B in Fun ( BG, C ∗ Alg nusep ). We next argue that one canfurther assume that p is a fibration. To this end we use the cylinder defined in (2.17) inorder to construct the commuting square A p (cid:47) (cid:47) e (cid:15) (cid:15) B Cyl( p ) q (cid:47) (cid:47) B Fun ( BG, C ∗ Alg nusep ), where q ( a, b ) = b (1) and e ( a ) = ( a, const p ( a ) ). The morphism e isa KK G -equivalence since it is an instance of the morphism u in (2.14). Therefore the twomorphisms kk G sep ( p ) and kk G sep ( q ) are equivalent.Note that q is surjective and that the exact cone sequence0 → C ( p ) → Cyl( p ) q → B → G sep is semiexact by Proposition 2.15.1, it sends this sequence to afibre sequence. This finishes the proof of Assertion 3.As a homotopy category of a stable ∞ -category, the category ho(KK G sep ) acquires atriangulated structure [Lur, Thm. 1.1.2.14]. On the other hand KK G has a triangulatedstructure described in [MN06, Sec. 2.1]. Proposition 2.18.
The functor ho : ho(KK G sep ) → KK G is an equivalence of triangulatedcategories.Proof. We know from Proposition 2.15.2 that ho is an equivalence of categories. Sincesums in ho(KK G sep ) and KK G are represented by sums in Fun ( BG, C ∗ Alg nusep ) we concludefurther that ho preserves sums and is therefore compatible with the Ab -enrichment.The inverse shift functor for KK G sep and therefore on ho(KK G sep ) is implemented by thesuspension functor (2.9) on the level of C ∗ -algebras via Lemma 2.6.2. Similarly, by thedescription given in [MN06, Sec. 2.1] the inverse shift functor on KK G is also implementedby (2.9) via Lemma 2.6.2. This shows that ho commutes with the shift functor.On the one hand, by [MN06, Sec. 2.1] the exact triangles in KK G are generated by themapping cone sequences for morphisms in Fun ( BG, C ∗ Alg nusep ). On the other hand, byLemma 2.17.3 the fibre sequences in KK G sep , and hence the exact triangles in ho(KK G sep )are also generated by the mapping cone sequences for morphisms in Fun ( BG, C ∗ Alg nusep ).We conclude that the functor ho is compatible with the collections of distinguishedtriangles.This finishes the proof of Theorem 1.3.KK G sep is stable by Proposition 2.15.2 and hence admits all finite colimits. The followingproposition strengthens this from finite to countable colimits and settles the remaining As-sertions 1.4.5 and 1.4.6. For the notion of an admissible diagram N → Fun ( BG, C ∗ Alg nusep )we refer to [MN06, Def. 2.5]. We will not repeat the definition here since the exact detailsare not relevant. We will only use its consequence [MN06, Prop. 2.6].32 emma 2.19.
1. The category KK G sep admits all countable colimits and is therefore idempotent complete.2. The functor kk G sep sends countable sums in Fun ( BG, C ∗ Alg nusep ) to coproducts.3. The functor kk G sep preserves colimits of all diagrams A : N → Fun ( BG, C ∗ Alg nusep ) which are admissible.Proof. Since KK G sep is stable, in order to show that KK G sep admits all countable colimits itsuffices to show that KK G sep admits countable coproducts. The functor kk G sep is essentiallysurjective by construction. Hence it suffices to show that for every countable family( A i ) i ∈ I in Fun ( BG, C ∗ Alg nusep ) the family (kk G sep ( A i )) i ∈ I in KK G sep admits a coproduct. Weconsider the sum A := (cid:76) i ∈ I A i in Fun ( BG, C ∗ Alg nusep ) and let e i : A i → A be the canonicalinclusion for every i in I . Then we claim that (kk G sep ( A ) , (kk G sep ( e i )) i ∈ I ) represents thecoproduct of the family (kk G sep ( A i ) i ∈ I ). In general, coproducts in a stable ∞ -category canbe detected on the homotopy category. In view of the equivalence kk G (cid:39) ho ◦ kk G sep givenby (1.2) it thus suffices to show that (kk G ( A ) , (kk G ( e i )) i ∈ I ) represents the coproduct inKK G . But this follows from [Kas88, Thm. 2.9] stating that the family of maps ( e i ) i ∈ I induces an isomorphism KK G ( A, B ) ∼ = −→ (cid:89) i ∈ I KK G ( A i , B )for every B in Fun ( BG, C ∗ Alg nusep ). The proof of Assertion 1 is finished with the generalobservation that a stable and countably cocomplete ∞ -category is idempotent complete.The claim also implies Assertion 2.We finally show the Assertion 3. Let A : N → Fun ( BG, C ∗ Alg nusep ) be an admissiblediagram. We must show that the canonical map colim N kk G sep ( A ) → kk G sep ( colim N A )is an equivalence. To this end it suffices to show that the map ho( colim N kk G sep ( A ) → ho(kk G sep ( colim N A )) obtained by composing with ho from (1.2) is an isomorphism.Using the isomorphismho( colim N kk G sep ( A )) ∼ = hocolim N ho(kk G sep ( A )) ∼ = hocolim N kk G ( A )and Proposition 2.18 (to translate homotopy colimits formed in the homotopy category ofthe stable ∞ -category KK G sep to homotopy colimits formed in the triangulated categoryKK G used in [MN06]), this is exactly the assertion of [MN06, Prop. 2.6].The existence of coproducts in KK G and the analogue of Assertion 2.19.2 for kk G haspreviously been shown in [MN06, Prop. 2.1].33he proof of Theorem 1.4 is now complete.In the following we consider the minimal and the maximal tensor products ⊗ min and ⊗ max of C ∗ -algebras. Both equip Fun ( BG, C ∗ Alg nusep ) with a symmetric monoidal structure.Recall that a symmetric monoidal structure on a stable ∞ -category is called bi-exact if thetensor product preserves cofibre sequences, and hence finite colimits, in each variable. Inthis case the ∞ -category together with its symmetric monoidal structure is called stablysymmetric monoidal. Proposition 2.20.
The tensor product ⊗ ? for ? in { min , max } descends to a bi-exactsymmetric monoidal structure on KK G sep and kk G sep refines to a symmetric monoidal functor kk G, ⊗ ? sep : Fun ( BG, C ∗ Alg nusep ) ⊗ ? → KK G, ⊗ ? sep . Proof.
In order to show that the tensor product descends to along the localizationkk G sep : Fun ( BG, C ∗ Alg nusep ) → KK G sep such that kk G sep refines to a symmetric monoidal functor we use [Hin16, Prop. 3.2.2]. Bythis result, it suffices to show that for every A in Fun ( BG, C ∗ Alg nusep ) the functor A ⊗ ? − preserves KK G -equivalences, where ? is in { min , max } . It is easy to see that the functorkk G ◦ ( A ⊗ ? − ) : Fun ( BG, C ∗ Alg nusep ) → KK G preserves zero objects and is K G -stable andsplit exact since kk G has these properties. We now apply Corollary 2.4 in order to concludethat this composition sends kk G -equivalences to isomorphisms. Hence A ⊗ ? − preserveskk G -equivalences.In order to show that the resulting symmetric monoidal structure on KK G sep is exactwe must show that kk G sep ( A ) ⊗ ? − in KK G sep preserves fibre sequences for every A in Fun ( BG, C ∗ Alg nusep ). We will use the observation made in the proof of Proposition 2.18that every fibre sequence in KK G sep is equivalent to a cone sequence.It suffices to show that kk G sep ( A ⊗ ? − ) sends cone sequences to fibre sequences.For the maximal tensor product, using the exactness of A ⊗ max − (see e.g. [BO08, Prop.3.7.1]) and the explicit description of the cone sequences in Remark 2.9 one checks that A ⊗ max − preserves cone sequences in Fun ( BG, C ∗ Alg nusep ). We then use that the functorkk G sep sends cone sequences to fibre sequences.For the minimal tensor product we use the fact that cone sequences admit completelypositive contractive splits and that the minimal tensor product is functorial for completelypositive contractive maps. We conclude that A ⊗ min − sends cone sequences to semisplitexact sequences. We now use that kk G sep is semixact.34 emark 2.21. Alternatively, the argument in the proof of Proposition 1.7 given for theminimal tensor product also applies to the maximal tensor product since the latter is alsofunctorial for completely positive contractive maps [Pis, Cor. 4.18].
Remark 2.22.
As a consequence of Proposition 2.20 the homotopy category ho(KK G sep )and therefore KK G has two tensor triangulated structures induced by ⊗ min and ⊗ max ,respectively. For ⊗ min this fact is well known, see e.g. [MN06, Sec. 2.5]. It seems thatthe symmetric monoidal structure coming from the maximal tensor product has not beenstudied so much, but for the non-equivariant case see e.g. [LN18, Lem. 3.13].Note that kk G sep is, by Definition 1.2, the initial functor from Fun ( BG, C ∗ Alg nusep ) to ∞ -categories which sends KK G -equivalences to equivalences. In Theorem 1.5 we have stateda different universal property which better reflects the standard properties of KK -theory.We will now first state an intermediate Theorem 2.23 involving additive ∞ -categories inorder to formulate a direct ∞ -categorial analog of [Mey00, Thm. 6.6] which appeared inthis paper as Proposition 2.2.Recall from [Lur, 6.1.6.13] that an ∞ -category is called semi-additive if it is pointed,admits finite products and coproducts, and if the canonical morphism from a coproduct tothe product of any two objects is an equivalence. The homotopy category of a semi-additivecategory is canonically enriched in abelian monoids. If all these morphism monoids areabelian groups then the ∞ -category is said to be additive. Stable ∞ -categories are additive.In particular, since KK G sep is stable by Proposition 2.15, it is additive. Theorem 2.23.
The functor kk G sep : Fun ( BG, C ∗ Alg nusep ) → KK G sep is initial among func-tors from Fun ( BG, C ∗ Alg nusep ) to objects of Cat add ∞ which are reduced, K G -stable, and splitexact.Proof. For an additive ∞ -category D we consider the full subcategory Fun rse ( Fun ( BG, C ∗ Cat nusep ) , D ) (2.30)of Fun ( Fun ( BG, C ∗ Cat nusep ) , D ) on functors which are reduced, K G -stable, and split exact.By Corollary 2.4 the functor category (2.30) is the full subcategory of the category Fun KK G ( Fun ( BG, C ∗ Alg nusep ) , D ) of functors sending KK G -equivalences to equivalences.We now build the following commutative diagram Fun (cid:96) (KK G sep , D ) Fun rse ( Fun ( BG, C ∗ Alg nusep ) , D ) Fun (KK G sep , D ) Fun KK G ( Fun ( BG, C ∗ Alg nusep ) , D ) (cid:39) (2.31)35he vertical morphisms are inclusions of full subcategories and the horizontal functors areinduced by precomposition with kk G sep . The superscript (cid:96) stands for coproduct preservingfunctors which are the morphisms in Cat add ∞ . By the universal property of kk G sep as aDwyer-Kan localization the lower horizontal functor is an equivalence. We now justifythat the dashed arrow exists, making the diagram commute. For this, we note thatthe localisation functor kk G sep is reduced, K G -stable and semiexact by Theorem 1.4. Inparticular it is split-exact. Thus if H : KK G sep → D preserves coproducts, the same is truefor the composition H ◦ kk G sep .In order to prove Theorem 2.23 we must show that the dashed arrow in (2.31) is anequivalence. As all other functors in diagram 2.31 are fully faithful, so is the dashed one.It remains to show that it is essentially surjective.To this end we consider F in Fun rhse ( Fun ( BG, C ∗ Alg nusep ) , D ). In view of the lower hori-zontal equivalence in (2.31) there exists a functor ¯ F in Fun (KK G sep , D ) and an equivalence¯ F ◦ kk G sep (cid:39) F . We must check that ¯ F ∈ Fun (cid:96) (KK G sep , D ), i.e., that ¯ F preserves coproducts.First of all F preserves the empty coproduct since kk G sep and F are reduced.We now show that ¯ F preserves binary coproducts. Since kk G sep and D are additive, inboth ∞ -categories coproducts and products coincide. Therefore it suffices to show that ¯ F preserves binary products. Since F is split-exact and reduced, it sends a split exact squareof the form A ⊕ B (cid:47) (cid:47) (cid:15) (cid:15) A (cid:15) (cid:15) B (cid:47) (cid:47) F preservesproducts. Now by Proposition 2.17.2 any product in KK G sep is equivalent to the image of aproduct in Fun ( BG, C ∗ Alg nusep ). In view of the equivalence ¯ F ◦ kk G sep (cid:39) F we can concludethat ¯ F preserves binary products.Hence ¯ F preserves binary coproducts. Since F preserves empty and binary coproducts itpreserves all finite coproducts.Theorem 2.23 says that if D is an object of Cat add ∞ and F : Fun ( BG, C ∗ Alg nusep ) → D is areduced, K G -stable, and split exact functor, then there exists an essentially unique, finitecoproduct-preserving factorization ¯ F as indicated in the diagram Fun ( BG, C ∗ Alg nusep ) F (cid:47) (cid:47) kk G sep (cid:15) (cid:15) D . KK G sep ¯ F (cid:55) (cid:55) D and KK G sep are additive, in both ∞ -categories products and coproducts coincide.Hence ¯ F also preserves all finite products. If D admits finite limits, e.g. if D is stable, onemight wonder whether the functor ¯ F in addition preserves finite limits. In general it doesnot, see Remark 2.25 for an example, but we have the following characterization.Let D be an additive ∞ -category and let F : Fun ( BG, C ∗ Alg nusep ) → D be a functor whichis reduced, K G -stable and split exact. Let ¯ F : KK G sep → D be the factorization as explainedabove. In the following statement we use the Definition 2.13.1 of semiexactness for functorswith additive targets. Theorem 2.24.
If in addition D admits finite limits and F is semiexact, then ¯ F preservesfinite limits.Proof. It suffices to prove that ¯ F is reduced and sends fibre sequences to fibre sequences.We already know that ¯ F is reduced by Theorem 2.23.By Proposition 2.17.3, any fibre sequence in KK G sep is the image under kk G sep of a semisplitexact sequence in Fun ( BG, C ∗ Alg nusep ). By assumption, F sends such a semisplit exactsequence to a fibre sequence. In view of the relation ¯ F ◦ kk G sep (cid:39) F we conclude that ¯ F sendsthe image of this semisplit exact sequence under kk G sep to a fibre sequence. Consequently¯ F preserves fibre sequences. Proof of Theorem 1.5.
The theorem immediately follows by specializing the Theorems2.23 and 2.24 to stable target categories.
Remark 2.25.
A natural example of a functor C ∗ Alg nusep → Sp which is reduced, K -stableand split exact is the composite C ∗ Alg nusep −→ Rings nuinv L −→ Sp where the first functor is the forgetful functor taking the underlying ring with involutionof a C ∗ -algebra and the second takes the (projective, symmetric) L-theory spectrum of aring with involution. This functor descends to a functor KK → Sp which preserves finiteproducts, but is not exact. Indeed, the failure of exactness in this case can be describedexplicitly, we refer to [LN18, Thm. 4.2] for a general treatment.
3. The s-finitary extension
Let F : Fun ( BG, C ∗ Alg nu ) → D be a functor to a target category D admitting all smallfiltered colimits. Then we have a canonical natural transformation ˆ F → F , where ˆ F is37he left Kan extension of F s := F | Fun ( BG,C ∗ Alg nusep ) as indicated in the following diagram: Fun ( BG, C ∗ Alg nusep ) F s (cid:47) (cid:47) incl (cid:15) (cid:15) D Fun ( BG, C ∗ Alg nu ) ⇓ ˆ F ⇓ (cid:52) (cid:52) F (cid:66) (cid:66) Lemma 3.1.
The functor F is s-finitary if and only if the natural transformation ˆ F → F is an equivalence.Proof. The pointwise formula for the left Kan-extension shows that colim ( A (cid:48) → A ) ∈ ( C ∗ Alg nusep ) /A F ( A (cid:48) ) (cid:39) ˆ F ( A ) . If we compare this formula with the morphism (1.14) appearing in the condition for beings-finitary it becomes clear that we must show that the poset of separable subalgebras of A is cofinal in ( C ∗ Alg nusep ) /A . This follows from the following observation.If f : A (cid:48) → A is any morphism in Fun ( BG, C ∗ Alg nu ), then we have a factorization A (cid:48) → f ( A (cid:48) ) → A , where f ( A (cid:48) ) is the G - C ∗ -subalgebra of A obtained by forming the closure of the image of f in A . If A (cid:48) is separable, then f ( A (cid:48) ) is separable, too.The next lemma is the essential step for the derivation of Theorem 1.17 from Theorem 1.4.Let F and F s be as above and assume in addition that D is stable. Lemma 3.2.
Assume that F is s-finitary.1. If F s is reduced, then so is F .2. If F s is homotopy invariant, then so is F .3. If F s is K G -stable, then so is F .4. If F s is semiexact, then so is F .5. If F s sends exact sequences to fibre sequences, then so does F .Proof. Assertion 1 is obvious since the zero algebra is separable.38e show Assertion 2. For every t in [0 ,
1] let ev t : C ([0 , → C be the evaluation at t .If B is an invariant separable subalgebra of C ([0 , ⊗ A , then the values ( ev t ⊗ id A )( b )for all b in B and t in [0 ,
1] generate an invariant separable subalgebra A (cid:48) of A suchthat B ⊆ C ([0 , ⊗ A (cid:48) . Hence the invariant subalgebras of C ([0 , ⊗ A the form C ([0 , ⊗ A (cid:48) with A (cid:48) an invariant separable subalgebra of A is cofinal in all invariantseparable subalgebras of C ([0 , ⊗ A . Since we assume that F is s-finitary we have thechain of equivalences F ( A ) (cid:39) colim A (cid:48) F s ( A (cid:48) ) (cid:39) colim A (cid:48) F s ( C ([0 , ⊗ A (cid:48) ) (cid:39) F ( C ([0 , ⊗ A ) , where the colimit runs over the poset of all invariant separable subalgebras A (cid:48) of A andthe middle equivalence is a consequence of the assumption on F s . Since this morphism isinduced by the canonical map (1.5) this shows that F is homotopy invariantWe show Assertion 3. Let A be in Fun ( BG, C ∗ Alg nu ), let H → H (cid:48) be an equivariantisometric inclusion of separable G -Hilbert spaces such that H (cid:54) = 0, and K ( H ) → K ( H (cid:48) )be the corresponding inclusion of the algebras of compact operators. We note that thesealgebras are separable. It follows that the family of G -invariant subalgebras A (cid:48) ⊗ K ( H )for all G -invariant separable subalgebras A (cid:48) of A is cofinal in all separable G -invariantsubalgebras of A ⊗ K ( H ), and similar for H (cid:48) . Using that F is s-finitary we concludethat the morphism obtained by applying F to (1.4) has a factorization over the chain ofequivalences F ( A ⊗ K ( H )) (cid:39) colim A (cid:48) F s ( A (cid:48) ⊗ K ( H )) (cid:39) colim A (cid:48) F s ( A (cid:48) ⊗ K ( H (cid:48) )) (cid:39) F ( A ⊗ K ( H )) , where the colimit runs over the poset of all invariant separable subalgebras A (cid:48) of A andthe middle equivalence follows from the assumption on F s .We show Assertion 4. We now consider a semisplit exact sequence0 → A (cid:48) i → A π → A (cid:48)(cid:48) → Fun ( BG, C ∗ Alg nu ). Let s : A (cid:48)(cid:48) → A denote the equivariant cpc split of π . We considerthe family S of exact sequences C : 0 → C (cid:48) → C → C (cid:48)(cid:48) → G -invariant subalgebras of (3.1) such that s ( C (cid:48)(cid:48) ) ⊆ C . Such a sequence isagain equivariantly semisplit since we can use the restriction of s . Since F is s-finitary itis enough to show that the families of constituents C (cid:48) , C , and C (cid:48)(cid:48) for C running in S arecofinal families of G -invariant separable C ∗ -algebras in A (cid:48) , A and A (cid:48)(cid:48) , respectively. Indeed,then using the assumption on F s the fibre sequence F ( A (cid:48) ) → F ( A ) → F ( A (cid:48)(cid:48) ) → Σ F ( A (cid:48) )39s the colimit over S of the family of fibre sequences F s ( C (cid:48) ) → F s ( C ) → F s ( C (cid:48)(cid:48) ) → Σ F s ( C (cid:48) ) . Let B (cid:48) , B and B (cid:48)(cid:48) be separable C ∗ -subalgebras of A (cid:48) , A , and A (cid:48)(cid:48) , respectively.Then the invariant C ∗ -subalgebra C (cid:48)(cid:48) := A (cid:48)(cid:48) ( π ( B ) , B (cid:48)(cid:48) ) of A (cid:48)(cid:48) generated by the subsets π ( B ) and B (cid:48)(cid:48) is separable and contains B (cid:48)(cid:48) . Then C := A ( B, s ( C (cid:48)(cid:48) ) , i ( B (cid:48) )) is a separablesubalgebra of A containing B such that the restriction of π induces a surjection C → C (cid:48)(cid:48) and s ( C (cid:48)(cid:48) ) ⊆ C . We finally define C (cid:48) := ker ( C → C (cid:48)(cid:48) ) considered as a subalgebra of A (cid:48) . Then B (cid:48) ⊆ C . Since B (cid:48) is a closed subspace of the separable C ∗ -algebra it is againseparable.By construction we have an equivariantly semisplit exact sequence0 → C (cid:48) → C → C (cid:48)(cid:48) → S and B (cid:48) ⊆ C (cid:48) , B ⊆ C , B (cid:48)(cid:48) ⊆ C (cid:48)(cid:48) . The proof of Assertion 5 is again a small modification of of the proof of Assertion 4. Sincewe do not have a split at all, we define C := A ( B, G ˆ C (cid:48)(cid:48) , i ( B (cid:48) )), where ˆ C (cid:48)(cid:48) is a set of choicesof preimages of a countable dense subset of C (cid:48)(cid:48) . Otherwise the argument is the same. Proof of Theorem 1.17.
Since the functor kk G is equivalent by Definition 1.10 to the leftKan-extension of its restriction kk G sep to separable G - C ∗ -algebras it is s-finitary by Lemma3.1. This is Assertion 1.17.1 which verifies the assumption in Lemma 3.2 for F = kk G and F s = kk G sep .The remaining assertions of Theorem 1.17 now follow by applying Lemma 3.2 and Theorem1.4 asserting the corresponding properties for kk G sep .Our next theorem settles the universal property of kk G stated as Theorem 1.18 in theintroduction. Theorem 3.3.
The functor kk G is initial among functors from Fun ( BG, C ∗ Alg nu ) toobjects of CAT ccpl ∩ colim ∞ which are s-finitary, reduced, K G -stable and semiexact.Proof. Let D be a cocomplete stable ∞ -category. We then have the following chain of40quivalences Fun colim (KK G , D ) y G, ∗ , (cid:39) → Fun ex (KK G sep , D ) (kk G sep ) ∗ , (cid:39) → Fun rse ( Fun ( BG, C ∗ Alg nusep ) , D ) incl ∗ , (cid:39) ← Fun frse ( Fun ( BG, C ∗ Alg nu ) , D ) , where the superscript colim stands for colimit preserving functors. Furthermore thesuperscript ( f ) rse stands the full subcategory of functors which are (s-finitary,) reduced, K G -stable and semiexact, respectively. .Here for the first two steps we use the universal properties of y G in (1.6) and kk G sep (Theorem1.5). The last equivalence is a consequence of the Lemmas 3.1 and 3.2 which imply that ifwe left Kan-extend a functor in Fun rse ( Fun ( BG, C ∗ Alg nusep ) , D ) along the inclusion incl in(1.7), then the result is in Fun frse ( Fun ( BG, C ∗ Alg nusep ) , D ). Since kk G ◦ incl (cid:39) y G ◦ kk G sep by (1.7) we conclude Fun colim (KK G , D ) (cid:39) −→ Fun frse ( Fun ( BG, C ∗ Alg nu ) , D )which is the desired equivalence. Remark 3.4.
Note that the functor y G : KK G sep → KK G in (1.6) only preserves finitecolimits. Therefore, in contrast to Proposition 2.19.2, the functor kk G only sends finitesums to coproducts. One could improve this situation by forming the Bousfieled localization L : KK G (cid:29) KK G, (cid:48) : inclof kk G at the set of maps (cid:76) n ∈ N kk G ( A n ) → kk G ( (cid:76) n ∈ N A n ) for all families ( A n ) n ∈ N in Fun ( BG, C ∗ Alg nusep ) and settingkk G, (cid:48) := L ◦ kk G : Fun ( BG, C ∗ Alg nu ) → KK G, (cid:48) . We restrict to families of separable algebras in order have a set of generators.The functor kk G, (cid:48) s-finitary. In addition, it preserves countable sums. Indeed, it preservescountable sums of families of separable G - C ∗ -algebras by construction. But using thatkk G, (cid:48) is s-finitary we can remove the assumption on separability.As an immediate consequence of Theorem 3.3 and the definitions the universal property ofkk G, (cid:48) is as follows: The functor kk G, (cid:48) is initial among functors from Fun ( BG, C ∗ Alg nu ) toobjects of CAT ccpl ∩ ex ∞ which are s-finitary, reduced, K G -stable, semiexact, and which sendcountable sums to coproducts. Note that in the proof of Theorem 2.23, the e in the superscript rse referred to split exact as opposedto semiexact in the present situation. We apologise for the notational clash. Proposition 3.5. If A is in Fun ( BG, C ∗ Alg nusep ) and B is in Fun ( BG, C ∗ Alg nu σ ) , thenwe have an isomorphism of Z -graded groups π ∗ KK G ( A, B ) ∼ = KK G ∗ ( A, B ) . (3.2) Proof.
Let A be in Fun ( BG, C ∗ Alg nusep ). We consider the functors π ∗ KK G ( A, − ) , KK G ∗ ( A, − ) : Fun ( BG, C ∗ Alg nu σ ) → Ab Z . (3.3)By the compatibility of ho with the triangulated structure stated in Theorem 1.3.2 itinduces a natural isomorphism π ∗ KK G ( A, ( − ) s ) ∼ = −→ KK G ∗ ( A, ( − ) s )between the restriction of these functors to separable G - C ∗ -algebras. It therefore suffices toshow that both functors in (3.3) are s-finitary. Since kk G ( A ) (cid:39) y G (kk G sep ( A )) is a compactobject of KK G and kk G is s -finitary, the functor KK G ( A, − ) is s-finitary, too. Since π ∗ preserves filtered colimits also π ∗ KK G ( A, − ) is s-finitary.In order to see that KK G ∗ ( A, − ) is s-finitary, using the shift equivalence of the triangulatedstructure on KK G , it suffices to show this for ∗ = 0. We now use the formula [Mey00,Thm. 5.5] (at this point we need the assumption that B is σ -unital) stating that KK G ( A, B ) ∼ = [ q s ( A ) ⊗ K, B ⊗ K ] . Here K := K ( (cid:96) ⊗ L ( G )), q s ( A ) := ker ( A ⊗ K (cid:116) A ⊗ K → A ⊗ K ), and [ − , − ] standsfor taking homotopy classes of morphisms in Fun ( BG, C ∗ Alg nu ). We note that K isseparable and q s ( A ) is separable. Every equivariant homomorphism q s ( A ) ⊗ K → B ⊗ K and every homotopy between such homomorphisms factorizes over an invariant separablesubalgebra of the target. Furthermore, the separable subalgebras of the form B (cid:48) ⊗ K forinvariant separable subalgebras B (cid:48) of B are cofinal. Consequently, KK G ( A, − ) is s-finitary,too.We now turn to the proof of Proposition 1.20 in the introduction (repeated here asProposition 3.8) concerning symmetric monoidal structures on kk G and KK G . As apreparation we show how left Kan extensions interact with symmetric monoidal structures.Let r : C → C (cid:48) be a symmetric monoidal functor between small symmetric monoidal ∞ -categories, and let D be a presentably symmetric monoidal ∞ -category. Lemma 3.6.
The left Kan extension r ! F : C (cid:48) → D of a lax symmetric monoidal functor F : C → D along r has a naturally induced lax symmetric monoidal refinement. roof. The restriction functor r ∗ : Fun ( C (cid:48) , D ) → Fun ( C , D ) is lax symmetric monoidal(with respect to the Day convolution structures on the functor categories). Furthermore ithas a symmetric monoidal left-adjoint r ! : Fun ( C , D ) → Fun ( C (cid:48) , D ) which is a symmetricmonoidal refinement of the left Kan-extension functor, see [Nik, Cor. 3.8]. This impliesthat r ! preserves commutative algebras.By [Gla16, Prop. 2.16], [Lur, Sec. 2.2.6] commutative algebras in these functor categoriescorrespond to lax symmetric monoidal functors. Therefore, if F : C → D is lax symmetricmonoidal, it is a commutative algebra in
Fun ( C , D ). Consequently, r ! F is a commutativealgebra in Fun ( C (cid:48) , D ), and hence r ! F is lax symmetric monoidal. Remark 3.7.
In proof of Proposition 3.8 below we will use that the Ind-completion Ind( D )of a symmetric monoidal ∞ -category D admits a canonical symmetric monoidal structurepreserving filtered colimits in each argument such that the canonical functor D →
Ind( D )has a canonical refinement to a symmetric monoidal functor. If D is stable and themonoidal structure on D is exact, then the induced structure on Ind( D ) is presentablysymmetric monoidal, and the canonical functor is symmetric monoidal. A reference forthese statements is [Lur, Cor. 4.1.8.14]. Proposition 3.8.
The symmetric monoidal structure ⊗ ? on KK G sep for ? in { min , max } in-duces a presentably symmetric monoidal structures on KK G and kk G refines to a symmetricmonoidal functor kk G, ⊗ ? : Fun ( BG, C ∗ Alg nu ) ⊗ ? → KK G, ⊗ ? . (3.4) Proof.
Applying Remark 3.7 to KK G sep with the symmetric monoidal structure ⊗ ? for ?in { min , max } constructed in Proposition 1.7 we get a presentably symmetric monoidalstructure ⊗ ? on KK G and an exact symmetric monoidal refinement y G, ⊗ ? : KK G, ⊗ ? sep → KK G, ⊗ ? of the functor in (1.6). Since kk G sep has a symmetric monoidal structure by Proposition 1.7,the upper horizontal composition in the diagram Fun ( BG, C ∗ Alg nusep ) kk G sep (cid:47) (cid:47) incl (cid:15) (cid:15) KK G sep y G (cid:47) (cid:47) KK G Fun ( BG, C ∗ Alg nu ) kk G (cid:52) (cid:52) has a symmetric monoidal structure, too. Applying Lemma 3.6 to the symmetric monoidalfunctor incl in place of r we conclude that kk G acquires a canonical lax symmetric monoidalrefinement (3.4).To show that kk G is in fact symmetric monoidal, we have to show that for A and B in Fun ( BG, C ∗ Alg nu ) the canonical mapkk G ( A ) ⊗ ? kk G ( B ) → kk G ( A ⊗ ? B ) (3.5)43s an equivalence. Furthermore, we must show that the unit morphism1 ⊗ ? KK G → kk G ( C ) (3.6)is an equivalence, where 1 ⊗ ? KK G is the tensor unit for the structure ⊗ ? on KK G , and C isthe tensor unit for Fun ( BG, C ∗ Alg nu ) for both structures.We start with the discussion of the unit morphism. By Remark 3.7 we have an equivalence1 ⊗ ? KK G (cid:39) y G (1 KK G sep ). By Proposition 1.7 we also know that 1 KK G sep (cid:39) kk G sep ( C ). If we nowapply y G to the second equivalence and compose with the first, then we get the desiredequivalence (3.5).We now give separate arguments for (3.5) in the cases where ? = max and ? = min.We first consider the case of ⊗ min . We will employ the following fact: If A and B are in C ∗ Alg nu and A (cid:48) is a closed subalgebra of A and B (cid:48) is a closed subalgebra of B (cid:48) , then thecanonical map A (cid:48) ⊗ min B (cid:48) → A ⊗ min B is an isometric inclusion.For completeness we provide an argument for this fact. Let A → B ( V ) and B → B ( W )be faithful representations of A and B on Hilbert spaces V and W , respectively. Then theminimal norm on A ⊗ min B can be defined by the tensor product representation of A ⊗ alg B on V ⊗ W . In order to define A (cid:48) ⊗ min B (cid:48) we can now use the faithful representations A (cid:48) → B ( V ) and B (cid:48) → B ( W ) induced by restriction. Then the minimal norm on A (cid:48) ⊗ alg B (cid:48) is the restriction of the minimal norm on A ⊗ alg B .The desired equivalence (3.5) for ⊗ min is given by the following chain of equivalences:kk G ( A ) ⊗ min kk G ( B ) (cid:39) colim A (cid:48) ⊆ A,B (cid:48) ⊆ B kk G sep ( A (cid:48) ) ⊗ min kk G sep ( B (cid:48) ) (cid:39) → colim A (cid:48) ⊆ A,B (cid:48) ⊆ B kk G sep ( A (cid:48) ⊗ min B (cid:48) ) (!) (cid:39) kk G ( A ⊗ min B ) . In the first step, we have used that kk G is s-finitary and that the tensor product in kk G commutes with colimits in each variable. In the second step we use that kk G sep is symmetricmonoidal for ⊗ min by Proposition 1.7. To justify the final equivalence (!), it suffices toshow that the poset of subalgebras of A ⊗ min B of the form A (cid:48) ⊗ min B (cid:48) (note that here weuse the fact from above), for separable subalgebras A (cid:48) and B (cid:48) of A and B , respectively,is cofinal in the poset of all separable subalgebras. To see this, let C be an invariantseparable subalgebra of A ⊗ min B . Let ( c i ) i ∈ I be a countable dense subset of C . For every i in I we can choose a countable family of elementary tensors ( a i,j ⊗ b i,j ) j ∈ J i such that c i belongs to the closure of the linear span of this family. We then let A (cid:48) be the subalgebragenerated by the G -orbits of the elements a i,j for all i in I and j in J i . It is an invariantseparable subalgebra of A . We define B (cid:48) similarly using the elements b i,j . By constructionwe have C ⊆ A (cid:48) ⊗ min B (cid:48) . This finishes the argument in the case of ⊗ min .44e now consider the maximal tensor product ⊗ max . The problem here is that if A (cid:48) and B (cid:48) are as above then we do not know that the canonical map A (cid:48) ⊗ max B (cid:48) → A ⊗ max B is anisometric inclusion. Denoting the image of this map by A (cid:48) ¯ ⊗ max B (cid:48) we get the surjection A (cid:48) ⊗ max B (cid:48) → A (cid:48) ¯ ⊗ max B (cid:48) .We use that A ∼ = colim A (cid:48) ⊆ A A (cid:48) , B ∼ = colim B (cid:48) ⊆ B B (cid:48) and the fact that the maximal tensorproduct commutes with filtered colimits in order to conclude that A ⊗ max B ∼ = colim A (cid:48) ⊆ A,B (cid:48) ⊆ B A (cid:48) ⊗ max B (cid:48) , (3.7)where as before the colimit runs over the poset of ( A (cid:48) , B (cid:48) ) of pairs of invariant separablesubalgebras of A and B , respectively. By a similar cofinality argument as in the case ofthe minimal tensor product we also get A ⊗ max B ∼ = colim A (cid:48) ⊆ A,B (cid:48) ⊆ B A (cid:48) ¯ ⊗ max B (cid:48) . (3.8)In the following we will show the claim that the inductive systems ( A (cid:48) ⊗ max B (cid:48) ) A (cid:48) ⊆ A,B (cid:48) ⊆ B and ( A (cid:48) ¯ ⊗ max B (cid:48) ) A (cid:48) ⊆ A,B (cid:48) ⊆ B are equivalent in Ind( C ∗ Alg nusep ).For the moment we fix A (cid:48) and B (cid:48) . We then get the outer part of the following diagram A (cid:48) ⊗ max B (cid:48) (cid:47) (cid:47) (cid:40) (cid:40) (cid:15) (cid:15) A (cid:48) ¯ ⊗ max B (cid:48) (cid:119) (cid:119) (cid:15) (cid:15) A (cid:48)(cid:48) ⊗ max B (cid:48)(cid:48) (cid:118) (cid:118) A ⊗ max B A ⊗ max B The vertical morphisms are the canonical inclusions into the colimits which have beenidentified using (3.7) and (3.8).Since A (cid:48) ⊗ max B (cid:48) is separable, the kernel I of A (cid:48) ⊗ max B (cid:48) → A (cid:48) ¯ ⊗ max B (cid:48) is separable. By thecommutativity of the square, it is annihilated by the left vertical map. By Example 7.16 theposet of separable subalgebras in a C ∗ -algebra is countably filtered . Since G is countable,using Lemma 7.17 we can find invariant separable subalgebras A (cid:48)(cid:48) and B (cid:48)(cid:48) containing A (cid:48) and B (cid:48) , respectively, such that I is annihilated by the map A (cid:48) ⊗ max B (cid:48) → A (cid:48)(cid:48) ⊗ max B (cid:48)(cid:48) .This provides the dotted morphism. The existence of ( A (cid:48)(cid:48) , B (cid:48)(cid:48) ) and the dotted arrow forany given ( A (cid:48) , B (cid:48) ) proves the claim. This means that every countable subset admits an upper bound. G ( A ) ⊗ max kk G ( B ) (cid:39) colim A (cid:48) ⊆ A,B (cid:48) ⊆ B kk G sep ( A (cid:48) ) ⊗ max kk G sep ( B (cid:48) ) (cid:39) → colim A (cid:48) ⊆ A,B (cid:48) ⊆ B kk G sep ( A (cid:48) ⊗ max B (cid:48) ) (!!) (cid:39) colim A (cid:48) ⊆ A,B (cid:48) ⊆ B kk G sep ( A (cid:48) ¯ ⊗ max B (cid:48) ) (!) (cid:39) kk G ( A ⊗ max B )providing the equivalence (3.5) in the case of the maximal tensor product. The remainingequivalences in this chain are justified in the same way as in the case of the minimal tensorproduct.
4. Change of groups functors
In this section we show that the restriction, induction and crossed product functors of C ∗ -algebras descend to functors between the corresponding universal KK-theoretic stable ∞ -categories. The following lemma is the blue-print for the assertions about the changeof groups functors below.Let G and H be countable groups and C : Fun ( BG, C ∗ Alg nu ) → Fun ( BH, C ∗ Alg nu ) besome functor and set C s := C | Fun ( BG,C ∗ Alg nusep ) . Lemma 4.1.
Assume:1. C preserves separable algebras.2. C is s-finitary.3. The composition kk H sep ◦ C s inverts KK G -equivalences.4. C s preserves semisplit exact sequences. hen there are functors F and F s taking part in the following commutative diagram KK G sep F s (cid:15) (cid:15) y G (cid:3) (cid:3) Fun ( BG, C ∗ Alg nusep ) kk G sep (cid:50) (cid:50) C s (cid:47) (cid:47) incl (cid:15) (cid:15) Fun ( BH, C ∗ Alg nusep ) incl (cid:15) (cid:15) kk H sep (cid:47) (cid:47) KK H sep y H (cid:15) (cid:15) Fun ( BG, C ∗ Alg nu ) kk G (cid:44) (cid:44) C (cid:47) (cid:47) Fun ( BH, C ∗ Alg nu ) kk H (cid:47) (cid:47) KK H KK GF (cid:79) (cid:79) (4.1) In addition:1. F s is exact.2. F preserves colimits and compact objects.Both F and F s are characterized by the diagram and these properties.Proof. The existence of the factorization C s and the commutativity of the left squarein (4.1) expresses the fact that C preserves separable algebras. The middle square is aninstance of (1.7).The existence of the factorization F s follows from Assumption 3 and the defining universalproperty of kk G sep as a Dwyer–Kan localization. Using Assumption 4 and the semiexactnessof kk H we conclude that kk H sep ◦ C s is semiexact, too. This implies by Theorem 2.24 that F s is exact.It follows from Assumption 2 and the fact that kk H is s-finitary that also kk H ◦ C is s-finitary.Using the existence of F s together with Proposition 3.2 we then verify that kk H ◦ C satisfiesthe assumptions of Theorem 3.3. We therefore obtain the colimit-preserving factorization F from the universal property of kk G .We claim that then also the right square commutes, i.e., that y H ◦ F s (cid:39) F ◦ y G . Inorder to see this we observe that the fillers of the other squares provide an equivalence y H ◦ F s ◦ kk G sep (cid:39) F ◦ y G ◦ kk G sep . Then the desired equivalence is again a consequence ofthe defining universal property of kk G sep as a Dwyer–Kan localization.The compact objects in KK G are given by the essential image of y G since KK G sep isidempotent complete by the Theorem 1.4.5. Therefore the commutativity of the rightsquare implies that F also preserves compact objects.47 emark 4.2. Note that F can alternatively be described as the functor Ind( F s ) obtainedby the applying the Ind-completion functor to F s .The fact that F preserves colimits and compact objects implies that F admits a right-adjoint which preserves filtered colimits. By stability it hence preserves all colimits andtherefore has a further right-adjoint. Below we will describe these right-adjoints explicitlyin some cases.Let H → G be a homomorphism of countable groups. The following lemma verifiesTheorem 1.21.1. Lemma 4.3.
We have the following commutative diagram KK G sepRes GH,s (cid:15) (cid:15) y G (cid:3) (cid:3) Fun ( BG, C ∗ Alg nusep ) kk G sep (cid:50) (cid:50) Res
GH,s (cid:47) (cid:47) incl (cid:15) (cid:15)
Fun ( BH, C ∗ Alg nusep ) incl (cid:15) (cid:15) kk H sep (cid:47) (cid:47) KK H sep y H (cid:15) (cid:15) Fun ( BG, C ∗ Alg nu ) kk G (cid:44) (cid:44) Res GH (cid:47) (cid:47) Fun ( BH, C ∗ Alg nu ) kk H (cid:47) (cid:47) KK H KK G Res GH (cid:79) (cid:79) (4.2) where1. Res
GH,s : KK G sep → KK H sep is exact.2. Res GH : KK G → KK H preserves colimits and compact objects.Proof. The assertions follow from Lemma 4.1 once we have verified its assumptions. It isobvious that Res GH preserves separable algebras and semisplit exact sequences, and that itis s-finitary. It is furthermore also obvious that the composition kk G sep ◦ Res
GH,s is reduced, K G -stable, and semiexact. By the Theorem 1.5 it therefore inverts KK G -equivalences. Remark 4.4.
From now one we will use the same notation Res GH for all restriction functorsinduced by H → G and hope that this does not produce confusion as the argument of thefunctor determines which version has to be considered. The same convention will lateralso apply to the functors Ind GH and − (cid:111) ? H considered below.The following corollary generalizes [MN06, (12)] from the triangulated to the stable ∞ -categorical level. 48 orollary 4.5. The functor
Res GH : KK G → KK H has a symmetric monoidal refinementfor both symmetric monoidal structures ⊗ min and ⊗ max .Proof. It suffices to prove that the functor Res
GH,s : KK G sep → KK H sep has a symmetricmonoidal refinement. The corollary then follows by passing to Ind-categories, sinceInd : Cat ex ∞ → Pr L st is a symmetric monoidal functor and hence induces a functor oncommutative algebras. To see the first assertion, we consider the diagram Fun ( BG, C ∗ Alg nusep ) Fun ( BH, C ∗ Alg nusep )KK G sep KK H sep F ¯ F and now use [Hin16, Prop. 3.2.2] together with Proposition 1.20. These results imply thatif F descends to a functor ¯ F as indicated, then ¯ F is symmetric monoidal if F is. Applyingthis to the symmetric monoidal functorRes GH,s : Fun ( BG, C ∗ Alg nusep ) → Fun ( BH, C ∗ Alg nusep )proves the corollary.We now assume that H is a subgroup of G . The following construction describes theinduction functor Ind GH . Construction 4.6.
For B in Fun ( BH, C ∗ Alg nu ) we let C b ( G, B ) denote the C ∗ -algebraof bounded B -valued functions on G with the sup-norm. The group G acts on C b ( G, B )by ( g, f ) (cid:55)→ ( g (cid:48) (cid:55)→ f ( g − g (cid:48) )). We define the induction functorInd GH : Fun ( BH, C ∗ Alg nu ) → Fun ( BG, C ∗ Alg nu )as follows:1. objects: The induction functor sends B in Fun ( BH, C ∗ Alg ) to the invariant subal-gebra Ind GH ( B ) of C b ( G, B ) generated by the functions f : G → B satisfyinga) f ( gh ) = h − f ( g ) for all h in H and g in G .b) The projection of supp ( f ) to G/H is finite.2. morphisms: The induction functor sends a morphism φ : B → B (cid:48) to the morphismInd GH ( φ ) : Ind GH ( B ) → Ind GH ( B (cid:48) ) given by Ind GH ( φ )( f )( g ) := φ ( f ( g )).We have a natural transformation of functors ι : id → Res GH ◦ Ind GH : Fun ( BH, C ∗ Alg nu ) → Fun ( BH, C ∗ Alg nu ) . (4.3)49ts evaluation at B in Fun ( BH, C ∗ Alg nu ) is given by the homomorphism ι B : B → Res GH (Ind GH ( B ))which sends b in B to the function G → B , g (cid:55)→ (cid:40) g − ( b ) if g ∈ H , g / ∈ H , on G .We frequently need the following well-known result. Let B be in Fun ( BH, C ∗ Alg nu ) and A be in Fun ( BG, C ∗ Alg nu ). Lemma 4.7.
For ? in { min , max } we have a canonical isomorphism Ind GH ( B ) ⊗ ? A ∼ = Ind GH ( B ⊗ ? Res GH ( A )) . (4.4) Proof.
Let ι B : Ind GH ( B ) → C b ( G, B ) denote the canonical inclusion. By construction ofthe induction functor we have the following commutative square in
Fun ( BG, C ∗ Alg nu ) C b ( G, B ) ⊗ ? A (cid:47) (cid:47) C b ( G, B ⊗ ? A )Ind GH ( B ) ⊗ ? A (cid:47) (cid:47) ι B ⊗ id A (cid:79) (cid:79) Ind GH ( B ⊗ ? Res GH ( A )) ι B ⊗ id A (cid:79) (cid:79) In order to show that the lower horizontal map is an isomorphism we choose a section r ofthe projection map G → G/H and extend the diagram non-equivariantly as follows: C b ( G, B ) ⊗ ? A (cid:47) (cid:47) r ∗ ⊗ id A (cid:15) (cid:15) C b ( G, B ⊗ ? A ) r ∗ (cid:15) (cid:15) C b ( G/H, B ) ⊗ ? A (cid:47) (cid:47) C b ( G/H, B ⊗ ? A ) C ( G/H, B ) ⊗ ? A (cid:79) (cid:79) ! (cid:47) (cid:47) C ( G/H, B ⊗ ? A ) (cid:79) (cid:79) Ind GH ( B ) ⊗ ? A ι B ⊗ id A (cid:55) (cid:55) ∼ = (cid:79) (cid:79) (cid:47) (cid:47) Ind GH ( B ⊗ ? Res GH ( A )) ι B ⊗ id A (cid:103) (cid:103) ∼ = (cid:79) (cid:79) The lower vertical isomorphisms are immediate consequences of the construction of theinduction functor. The arrow marked by ! is an isomorphism because of C ( G/H, B ) ⊗ ? A ∼ = C ( G/H ) ⊗ ? B ⊗ ? A ∼ = C ( G/H, B ⊗ ? A ) . Hence the lower horizontal morphism is an isomorphism, too.50he following lemma verifies Theorem 1.21.2.
Lemma 4.8.
We have the following commutative diagram KK H sepInd GH,s (cid:15) (cid:15) y H (cid:3) (cid:3) Fun ( BH, C ∗ Alg nusep ) kk H sep (cid:50) (cid:50) Ind
GH,s (cid:47) (cid:47) incl (cid:15) (cid:15)
Fun ( BG, C ∗ Alg nusep ) incl (cid:15) (cid:15) kk H sep (cid:47) (cid:47) KK G sep y G (cid:15) (cid:15) Fun ( BH, C ∗ Alg nu ) kk H (cid:44) (cid:44) Ind GH (cid:47) (cid:47) Fun ( BG, C ∗ Alg nu ) kk H (cid:47) (cid:47) KK G KK H Ind GH (cid:79) (cid:79) (4.5) where1. Ind
GH,s : KK H sep → KK G sep is exact.2. Ind GH : KK H → KK G preserves colimits and compact objects.Proof. The assertion will again follow from Lemma 4.1. In comparison with the case ofRes GH the verification of Assumption 4.1.3 is considerably more complicated. It will employnon-formal results of Kasparov [Kas88] which we will use in the form stated in [MN06](see Remark 4.9).It is easy to see that Ind GH preserves separable algebras and is s-finitary. Here we use thatour groups are countable.Further, Ind GH preserves semisplit exact sequences. To this end let p : A → B be a surjectivemorphism in Fun ( BH, C ∗ Alg nu ) and s : B → A be an equivariant completely positivecontraction such that p ◦ s = id B . Then we define (using notation from Construction 4.6)a map of vector spaces Ind GH ( s ) : Ind GH ( B ) → Ind GH ( A ) by Ind GH ( s )( f )( g ) := s ( f ( g )). Thismap preserves the generators (i.e., functions satisfying the Conditions 4.6.1a and 4.6.1b)by the linearity, continuity and equivariance of s . We now show that Ind GH ( s ) extends bycontinuity to a completely positive contraction. We choose a section of the projection map G → G/H . The restriction along this section induces an isomorphism of C ∗ -algebrasRes G (Ind GH ( B )) ∼ = C ( G/H ) ⊗ min Res H ( B )and similarly for A . Under this identification the map Res G (Ind GH ( s )) acts as id C ( G/H ) ⊗ s .This map extends by continuity to a completely positive contraction. For this last step weuse that completely positive contractions are compatible with minimal tensor products.51e now show that the composition kk G sep ◦ Ind
GH,s : Fun ( BH, C ∗ Alg nusep ) → KK G sep invertsKK H -equivalences. We give a short argument using results from the literature. Using theadjunction [MN06, (20)] and the Yoneda lemma we see that the compositionho ◦ kk G sep ◦ Ind
GH,s : Fun ( BH, C ∗ Alg nusep ) → KK G sends KK H -equivalences to isomorphisms. Since ho detects equivalences we hence concludethat kk G ◦ Ind
GH,s sends KK H -equivalences to equivalences. Remark 4.9.
The details of the arguments leading to [MN06, (20)] are not very welldocumented in the literature. Therefore we sketch now an alternative argument for thefact that kk G sep ◦ Ind
GH,s : Fun ( BH, C ∗ Alg nusep ) → KK G sep inverts KK H -equivalences which isclose to the general philosophy of the present paper. We consider the G -Hilbert spaces V G := L ( G ) and V (cid:48) G := C ⊕ V G , where C has the trivial G -action. We define (cid:102) Ind GH : Fun ( BH, C ∗ Alg nu ) → Fun ( BG, C ∗ Alg nu )by (cid:102) Ind GH ( B ) := Ind GH ( B ⊗ min Res GH K ( V G )) . (4.6)Since K ( V G ) is separable, the functor (cid:102) Ind GH preserves separable algebras. As before weindicate the restriction of an induction functor to separable algebras by a subscript ‘s’, butin order to simplify the notation we drop this subscript at restriction functors. We claim:1. kk G sep ◦ (cid:102) Ind
GH,s sends KK H -equivalences to equivalences.2. We have an equivalence kk G sep ◦ Ind
GH,s (cid:39) kk G sep ◦ (cid:102) Ind
GH,s .Both assertions together imply that kk G sep ◦ Ind
GH,s inverts KK H -equivalences.We first show Assertion 2 of the claim as follows. We consider the diagram B → B ⊗ min Res GH K ( V (cid:48) G ) ← B ⊗ min Res GH K ( V G )where the maps are induced by the obvious G -equivariant isometric embeddings C → V (cid:48) G and V G → V (cid:48) G . We now assume that B is separable, apply kk G sep ◦ Ind
GH,s and use (4.4) inorder to getkk G sep (Ind GH,s ( B )) → kk G sep (Ind GH,s ( B ) ⊗ min K ( V (cid:48) G )) ← kk G sep (Ind GH,s ( B ) ⊗ min K ( V G )) def. = kk G sep ( (cid:102) Ind
GH,s ( B )) . By K G -stability of kk G sep , the first two arrows are equivalences. The whole construction isnatural in B and provides the equivalence claimed in Assertion 2. The text before (9) in [MN06, (20)] suggests that the authors only consider K -stability while one needs K G -stability in order to apply the universal property from Corollary 2.4.
52n order to show Assertion 1 we show thatkk G sep ◦ (cid:102) Ind
GH,s : Fun ( BH, C ∗ Alg nusep ) → KK G sep (4.7)is reduced, K H -stable and semiexact. Then we apply the universal property of kk H sep statedin Theorem 1.5.Since (cid:102) Ind
GH,s (0) = 0 and kk G sep is reduced, the functor in (4.7) is reduced as well.Since the operations − ⊗ min K ( V G ) and Ind GH,s (as seen in the proof of Lemma 4.8) preservesemisplit exact sequences we conclude that (cid:102)
Ind
GH,s also preserves semisplit exact sequences.Since kk G sep is semiexact we conclude that (4.7) is semiexact.The most complicated part of the argument is K H -stability. Let V → V (cid:48) be a unitaryinclusion of non-trivial H -Hilbert spaces and B be in Fun ( BG, C ∗ Alg nusep ). We must showthat the induced mapkk G sep ( (cid:102) Ind
GH,s ( B ⊗ min K ( V ))) → kk G sep ( (cid:102) Ind
GH,s ( B ⊗ min K ( V (cid:48) ))) (4.8)is an equivalence.If W is any H -Hilbert space, then we have an isomorphism of H -Hilbert spaces W ⊗ L ( H ) ∼ = −→ Res H W ⊗ L ( H ) , w ⊗ [ h ] (cid:55)→ hw ⊗ [ h ] , (4.9)where [ h ] in L ( H ) denotes the basis element corresponding to h in H . Here H actsdiagonally on the left hand side, and only on the factor L ( H ) on the right-hand sideDecomposing G into H -orbits we obtain an H -equivariant isomorphismRes GH ( L ( G )) ∼ = L ( H ) ⊗ L ( G/H ) , (4.10)where H acts by left-translations on the first tensor factor and trivially on the second.Combining (4.9) and (4.10) we get an isomorphism of H -Hilbert spaces W ⊗ Res GH ( L ( G )) (4.10) ∼ = W ⊗ L ( H ) ⊗ L ( G/H ) (4.9) ∼ = Res H W ⊗ L ( H ) ⊗ L ( G/H ) (4.10) ∼ = Res GH (Res H W ⊗ L ( G )) . This isomorphism is natural in the H -Hilbert space W .We conclude that the homomorphism of H -algebras K ( V ) ⊗ min Res GH ( K ( L ( G ))) → K ( V (cid:48) ) ⊗ min Res GH ( K ( L ( G )))is isomorphic to the homomorphismRes GH K (Res H V ⊗ L ( G ))) → Res GH K (Res H V (cid:48) ⊗ L ( G ))) .
53n view of (4.6) this implies that the map (cid:102)
Ind
GH,s ( B ⊗ min K ( V )) → (cid:102) Ind
GH,s ( B ⊗ min K ( V (cid:48) ))is isomorphic to the mapInd GH,s ( B ⊗ min Res GH K (Res H V ⊗ L ( G ))) → Ind
GH,s ( B ⊗ min Res GH K (Res H V (cid:48) ⊗ L ( G ))) . Applying (4.4) we furthermore see that the latter is isomorphic toInd
GH,s ( B ) ⊗ min K (Res H V ⊗ L ( G ))) → Ind
GH,s ( B ) ⊗ min K (Res H V (cid:48) ⊗ L ( G ))) . Since the functor kk G sep is K G -stable, it sends this map to an equivalence. Consequently,(4.8) is an equivalence.The following corollary generalizes the projection formula [MN06, (16)] from the triangu-lated to the ∞ -categorical level. Corollary 4.10.
For ? in { min , max } we have an equivalence of functors Ind GH ( − ) ⊗ ? ( − ) (cid:39) Ind GH (( − ) ⊗ ? Res GH ( − )) : KK H × KK G → KK G . Proof.
This is an immediate consequence of Lemma 4.7, Lemma 4.3, Lemma 4.8 andProposition 1.20.For A in Fun ( BG, C ∗ Alg nu ) we can form the maximal and reduced crossed products A (cid:111) max G and A (cid:111) r G in C ∗ Alg nu . In the arguments below we need some details of theirconstruction which we will therefore recall at this point. Construction 4.11.
Both crossed products are defined as completions of the algebraiccrossed product A (cid:111) alg G . The latter is the ∗ -algebra generated by elements ( a, g ) with a in A and g in G with multiplication ( a (cid:48) , g (cid:48) )( a, g ) := ( g − ( a (cid:48) ) a, g (cid:48) g ) and the involution( a, g ) ∗ := ( g ( a ∗ ) , g − ) subject to the relations ( a, g ) + λ ( a (cid:48) , g ) = ( a + λa (cid:48) , g ) for all a, a (cid:48) in A , λ in C and g in G . The maximal crossed product A (cid:111) max G is the completion of A (cid:111) alg G inthe maximal norm, and the reduced crossed product A (cid:111) r G is the completion of A (cid:111) alg G in the norm induced by the canonical representation on the A -Hilbert C ∗ -module L ( G, A ),see e.g. [BEa, (10.10)] for an explicit formula.If H is a subgroup of G , then we have a canonical homomorphismRes GH ( A ) (cid:111) ? H → A (cid:111) ? G (4.11)given on generators by ( a, h ) (cid:55)→ ( a, h ), where h in the target is considered as an elementof G . 54he next lemma provides the remaining Assertion 3 of Theorem 1.21. Lemma 4.12.
We have the following commuting diagram KK G sep( − (cid:111) ? G ) s (cid:15) (cid:15) y G (cid:3) (cid:3) Fun ( BG, C ∗ Alg nusep ) kk G sep (cid:50) (cid:50) ( − (cid:111) ? G ) s (cid:47) (cid:47) incl (cid:15) (cid:15) C ∗ Alg nusepincl (cid:15) (cid:15) kk sep (cid:47) (cid:47) KK sep y (cid:15) (cid:15) Fun ( BG, C ∗ Alg nu ) kk G (cid:44) (cid:44) − (cid:111) ? G (cid:47) (cid:47) C ∗ Alg nu kk (cid:47) (cid:47)
KKKK G − (cid:111) ? G (cid:79) (cid:79) (4.12) where1. ( − (cid:111) ? G ) s : KK G sep → KK sep is exact.2. − (cid:111) ? G : KK G → KK preserves colimits and compact objects.Proof. We explain the details of the argument in the case of the maximal crossed productand indicate where modifications are needed in the case of the reduced crossed product.The assertion will follow from Lemma 4.1. It is easy to check that the functor − (cid:111) max G preserves separable algebras and is s-finitary. Moreover, the restriction ( − (cid:111) max G ) s pre-serves semisplit exact sequences by [Mey08, Prop. 9]. Note that the latter is also true forthe reduced crossed product by the same reference.In order to verify Assumption 4.1.3 we will show that the functorkk sep ◦ ( − (cid:111) max G ) s : Fun ( BG, C ∗ Alg nusep ) → KK sep (4.13)is reduced, K G -stable and semiexact. It then follows from the universal property stated inTheorem 1.5 that it inverts KK G -equivalences.The functor (4.13) and its variant for (cid:111) r are clearly reduced.If B is in C ∗ Alg nu and A is in Fun ( BG, C ∗ Alg nu ), then we have a canonical isomorphism( A (cid:111) ? G ) ⊗ ! B ∼ = ( A ⊗ ? B ) (cid:111) ! G , (4.14)where (? , !) ∈ { (max , max) , (min , r ) } . On elementary tensors it is induced by the map( a, h ) ⊗ b (cid:55)→ ( a ⊗ b, h ). For a proof in the case (? , !) = (max , max) (also stated in [Mey08,(3)]) we refer to [Wil07, Lem. 2.75]. For the case (? , !) = (min , r ) see [Ech10, Lem. 4.1]. In55oth cases the argument is similar to the proof of Lemma 4.13 below. The isomorphism(4.14) could be used to show K -stability, but it is not sufficient to show that (4.13) is K G -stable since there we would need it for algebras of the form K ( V ) in place of B whichhave a non-trivial G -action. This problem is settled by the following lemma which isprobably well-known to the experts. Lemma 4.13. If V is a separable G -Hilbert space, then for ? ∈ { max , r } we have acanonical isomorphism ( A ⊗ K ( V )) (cid:111) ? G ∼ = ( A (cid:111) ? G ) ⊗ K ( V ) . Proof.
Note that K ( V ) and Res G ( K ( V )) denote the same thing. Since K ( V ) is nuclear wecan omit the decoration min or max at the tensor products. As a first step we constructan isomorphism φ : ( A ⊗ alg K ( V )) (cid:111) alg G (cid:39) −→ ( A (cid:111) alg G ) ⊗ alg K ( V ) . We define φ (( a ⊗ k, h )) := ( a, h ) ⊗ hk . It is straightforward to check that φ is a homomorphism and compatible with the involution.The inverse of φ is given by ψ : ( A (cid:111) alg G ) ⊗ alg K ( V ) (cid:39) −→ ( A ⊗ alg K ( V )) (cid:111) alg G , ψ (( a, h ) ⊗ k ) := ( a ⊗ h − k, h ) . We must show that these isomorphisms extend to the corresponding completions. We firstdiscuss the case of the maximal crossed product.Assume that we have a homomorphism of ∗ -algebras ( A ⊗ alg K ( V )) (cid:111) alg G → B with B a C ∗ -algebra. We can consider A ⊗ alg K ( V ) naturally as a subalgebra of ( A ⊗ alg K ( V )) (cid:111) alg G by a ⊗ h (cid:55)→ ( a ⊗ h, e ). The restriction of the representation above extends to A ⊗ K ( V )and then further to ( A ⊗ K ( V )) (cid:111) alg G , and hence to ( A ⊗ K ( V )) (cid:111) G .Similarly, consider a homomorphism of ∗ -algebras ( A (cid:111) alg G ) ⊗ alg K ( V ) → B . For everyfinite-dimensional matrix subalgebra E in K ( V ) we get a representation of ( A (cid:111) alg G ) ⊗ E by restriction along ( A (cid:111) alg G ) ⊗ alg E → ( A (cid:111) alg G ) ⊗ alg K ( V ). These representationsextend to ( A (cid:111) G ) ⊗ E . Since every operator in K ( V ) can be approximated by finite-dimensional ones we can further extend the representation above to a representation of( A (cid:111) G ) ⊗ alg K ( V ), and hence again to ( A (cid:111) G ) ⊗ K ( V ).The two observations above show that φ and ψ are isometries with respect to the maximalnorms. This finishes the case of the maximal crossed product.For the reduced crossed product we interpret ⊗ as the minimal, i.e. spatial, tensor product.The norm on ( A (cid:111) alg G ) ⊗ alg K ( V ) is induced from the representation on L ( G, A ) ⊗ V ,and the norm on ( A ⊗ alg K ( V )) (cid:111) alg G is induced from the representation on L ( G, A ⊗ V ).56e have an isometry U : L ( G, A ) ⊗ V → L ( G, A ⊗ V ) which sends α ⊗ v for α in L ( G, A )and v in V to the function h (cid:55)→ α ( h ) ⊗ h − v . The isometry U intertwines the representationof ( A (cid:111) alg G ) ⊗ alg K ( V ) on L ( G, A ) ⊗ V with its representation via ψ on L ( G, A ⊗ V ).Hence ψ is isometric for the reduced crossed product and minimal tensor product. Thisfinishes the argument in the case of the reduced crossed product.We resume with the proof of Lemma 4.12. We have seen that the functor (4.13) is reduced.Using Lemma 4.13 we see that it is K G -stable. The functor − (cid:111) max G sends semisplitexact sequences in Fun ( BG, C ∗ Alg nusep ) to semisplit exact sequences in C ∗ Alg nusep [Mey08,Prop. 9]. By the same reference this is also correct for the version − (cid:111) r G . Using thatkk sep is semiexact we conclude that (4.13) is semiexact in both cases. As argued abovethese four properties verify Assumption 4.1.3.Thus we have completed the proof of Theorem 1.21 from the introduction.The existence of a right-adjoint of Res GH : KK G → KK H for abstract reasons has beenobserved in Remark 4.2. In the following we identify it explicitly with a functor inducedby a functor on the algebra level provided H is a subgroup of G of finite index. Construction 4.14.
Let H be any subgroup of G . We define the coinduction functorCoind GH : Fun ( BH, C ∗ Alg nu ) → Fun ( BG, C ∗ Alg nu ) , A (cid:55)→ Coind GH ( A ) := C b ( G, A ) H , where C b ( G, A ) H is the subspace of the algebra C b ( G, A ) of bounded functions f : G → A which are H -invariant in the sense that f ( gh ) = h − f ( g ) for all g in G and h in H . The ∗ -algebra structure on C b ( G, A ) H is pointwise induced by the ∗ -algebra structure on A ,and the G -action is by left-translation of functions. Note that C b ( G, A ) H is not separable,in general. We furthermore define a natural transformationRes GH ◦ Coind GH → id (4.15)whose value at A is the H -equivariant evaluation map C b ( G, A ) H (cid:51) f (cid:55)→ f ( e ) ∈ A .It is straightforward to check that (4.15) is a counit of an adjunctionRes GH : Fun ( BG, C ∗ Alg nu ) (cid:29) Fun ( BH, C ∗ Alg nu ) : Coind GH . (4.16) Proposition 4.15. If H is a subgroup of finite index in G , then the adjunction (4.16) descends to an adjunction Res GH : KK G (cid:29) KK H : Coind GH . (4.17)57 roof. By Lemma 4.3 we know that Res GH descends to the stable ∞ -categories. In orderto show the proposition it therefore suffices to show that Coind GH also descends. But thisfollows from Lemma 4.8 since we have an isomorphism Ind GH ∼ = Coind GH for subgroups H of finite index. Remark 4.16.
In this remark we explain why it is not clear whether Proposition 4.15holds true in the case when the index of the subgroup H of G is not finite. We would liketo show that kk G ◦ Coind GH : Fun ( BH, C ∗ Alg nu ) → KK G is s -finitary, reduced, K H -stable and split exact. The properties split exact, reduced and s -finitary are straightforward. The problematic property is K H -stability. For example, inorder to verify the special case of K -stability, we would like to use that the canonical map K ( (cid:96) ) ⊗ C b ( G, A ) H → C b ( G, K ( (cid:96) ) ⊗ A ) H is an isomorphism. But this is wrong, see e.g. [Wil03] for a detailed discussion. In theverification of K G -stablity we would need the analog of Lemma 4.7 for Coind GH in place ofInd GH which fails for a similar reason.The natural transformation (4.3) descends to a natural transformation of functorskk G ( ι ) : id → Res GH ◦ Ind GH : KK H → KK H . (4.18)The following proposition settles Assertions 1 and 2 of Theorem 1.22. Proposition 4.17.
1. The transformation kk G ( ι ) is the unit of an adjunction Ind GH : KK H (cid:29) KK G : Res GH .
2. The transformation (1.17) naturally induces an equivalence of functors ( − ) (cid:111) ? H → Ind GH ( − ) (cid:111) ? G : KK H → KK for ? in { r, max } .Proof. For Assertion 1 we must show that the natural transformation of functors (KK H ) op × KK G → Sp r GH : KK G (Ind GH ( − ) , − ) Res GH −−−→ KK H (Res GH (Ind GH ( − )) , Res GH ( − )) (4.19) (4.18) −−−→ KK H ( − , Res GH ( − )) .
58s an equivalence. The domain and the target of (4.19) send filtered colimits in KK H tolimits in Sp . Since KK H is generated by KK H sep under filtered colimits it suffices to showthat the restriction of (4.19) to (KK H sep ) op × KK G is an equivalence. Since now both sidespreserve filtered colimits in KK G it suffices to consider the restriction to (KK H sep ) op × KK G sep .Finally, since both sides are compatible with fibre sequences it actually suffices to showthat the transformation π r GH : π KK G sep (Ind GH ( − ) , − ) → π KK H sep ( − , Res GH ( − ))of group-valued functors on (KK H sep ) op × KK G sep obtained by applying π is an isomorphism.But this is the same as that in [MN06, (20)], which is shown to be an isomorphism inloc. cit. This finishes the proof that (4.19) is an equivalence and hence of Assertion 1.To see the Assertion 2 first observe that the domain and the target of the transformation ares-finitary. Hence it suffices to check the equivalence after restriction to Fun ( BH, C ∗ Alg nusep ).We now use Green’s Imprimitivity Theorem which states that for A in Fun ( BH, C ∗ Alg nusep )(and also more general H - C ∗ -algebras) the map of C ∗ -algebras A (cid:111) ? H → Ind GH ( A ) (cid:111) ? G induced by (1.17) is a Morita equivalence and hence a kk -equivalence. We finally usethat kk inverts kk -equivalences. Remark 4.18.
The Proposition 4.17.1 identifies the right adjoint of Ind GH : KK H → KK G whose existence was prediced by Remark 4.2 with Res GH : KK G → KK H .Recall the internal morphism object functor kk G ? ( − , − ) from (1.15). Corollary 4.19.
For ? in { min , max } we have an equivalence of functors Res GH ◦ kk G ? ( − , − ) (cid:39) kk H ? (Res GH ( − ) , Res GH ( − )) : (KK G ) op × KK G → KK H . (4.20) Proof.
This is a formal consequence of the adjunction in Proposition 4.17.1 together withCorollary 4.10.We assume that H is a finite group. If A is in C ∗ Alg nu , then we consider the homomor-phism (cid:15) A : A → Res H ( A ) (cid:111) H , a (cid:55)→ | H | (cid:88) h ∈ H ( a, h ) . The text before [MN06, (9)] suggests that the authors wanted to state the version of Green’s ImprimitivityTheorem on the level of their triangulated categories. This version follows from our ∞ -categoricalversion by going to the homotopy category. Our proof is essentially the same argument as envisaged byMeyer–Nest for the justification of [MN06, (9)] with the crucial addendum, that the Morita equivalenceis induced by a morphism on the level of C ∗ -algebras. H ( A ) denotes the C ∗ -algebra A equipped with the trivial H -action. Thefamily (cid:15) = ( (cid:15) A ) A ∈ C ∗ Alg nu is a natural transformation (cid:15) : id → Res H ( − ) (cid:111) H (4.21)of endofunctors of C ∗ Alg nu . It gives rise to a natural transformationGJ H : KK H (Res H ( − ) , − ) − (cid:111) H −−−→ KK(Res H ( − ) (cid:111) H, − (cid:111) H ) kk( (cid:15) ) ∗ −−−→ KK( − , − (cid:111) H ) (4.22)of functors from KK op × KK H to Sp . Here GJ H stands for Green–Julg.The first assertion of the following theorem is a spectrum-level generalization of theclassical Green–Julg theorem. For a finite group H it explicitly identifies the right-adjointof Res H : KK → KK H whose existence was predicted by Remark 4.2. Recall that G isalways assumed to be a countable group. Theorem 4.20.
Let H be a finite group. Then the transformation (4.21) induces theunit of an adjunction Res H ( − ) : KK (cid:29) KK H : − (cid:111) H . (4.23)
Proof.
We must show that (4.22) is an equivalence. Since the functors in the domain andthe target of (4.22) send filtered colimits in KK to limits and KK is generated by KK sep under filtered colimits it suffices to show that the restriction of (4.22) to KK opsep × KK H is an equivalence. Since this restriction preserves filtered colimits in KK H it suffices toconsider the restriction to KK opsep × KK H sep . Since both sides are compatible with suspensionsit suffices to show that we get an isomorphism of group-valued functors after applying π .It thus suffices to show that we get an isomorphism KK H (Res H ( A ) , B ) ∼ = −→ KK ( A, B (cid:111) H ) (4.24)for every A in C ∗ Alg nusep and B in Fun ( BG, C ∗ Alg nusep ). This is the classical Green–Julgtheorem (see [Bla98, Thm. 20.2.7] for the case A = C ).In the following we sketch the argument. For KK H we work with Kasparov (Res H ( A ) , B )-modules (( M, ρ ) , φ, F ) such that F is H -equivariant, i.e. ρ ( h ) F = F ρ ( h ) for all h in H .Here ρ ( h ) : M → M denotes the action of h in H on M such that (cid:104) ρ ( h )( m ) , m (cid:48) (cid:105) = h ( (cid:104) m, ρ ( h − )( m (cid:48) ) (cid:105) )for all m, m (cid:48) in M . If H acts non-trivially on B , then ρ ( h ) is not right- B -linear in general.Following the definitions the map (4.24) sends the (Res H ( A ) , B )-module (( M, ρ ) , φ, F ) tothe ( A, B (cid:111) H )-module ( M (cid:111) H, ( φ (cid:111) H ) ◦ (cid:15) A , F (cid:111) H ) . (4.25)Here M (cid:111) H is the Hilbert ( B (cid:111) H )-module whose underlying C -vector space is givenby (cid:76) h ∈ H M . We write the generators, which have only one non-zero entry m in the60ummand indexed by h , in the form ( m, h ). Then the right-action of B (cid:111) H is givenby ( m, h )( b, h (cid:48) ) = ( ρ ( h (cid:48) , − )( m ) b, hh (cid:48) ). Furthermore, the ( B (cid:111) H )-valued scalar productis given by (cid:104) ( m, h ) , ( m (cid:48) , h (cid:48) ) (cid:105) := ( (cid:104) ρ ( h (cid:48) , − h )( m ) , m (cid:48) (cid:105) E , h − h (cid:48) ). The operator F (cid:111) H acts as F ( m, h ) = ( F ( m ) , h ). Finally, the representation φ (cid:111) H of A (cid:111) H on M (cid:111) H is given by( φ (cid:111) H )( a, h )( m, h (cid:48) ) = ( φ ( a )( m ) , hh (cid:48) ).In the following we find a simpler representative of the class of (4.25) which is more directlyrelated with (( M, ρ ) , φ, F ). We consider the projection π ∈ B ( M (cid:111) H ) given by π ( m, h ) := 1 H (cid:88) h (cid:48) ∈ H ( ρ ( h (cid:48) )( m ) , h (cid:48) h ) . This projection commutes with F (cid:111) H and ( φ (cid:111) H )( (cid:15) A ( a )) for every a in A . Furthermore, π ◦ ( φ (cid:111) H )( (cid:15) A ( a )) = ( φ (cid:111) H )( (cid:15) A ( a )) and (1 − π ) ◦ ( φ (cid:111) H )( (cid:15) A ( a )) = 0for all a in A . We decompose( M (cid:111) H, ( φ (cid:111) H ) ◦ (cid:15) A , F (cid:111) H ) ∼ = ( π ( M (cid:111) H ) , ( φ (cid:111) H ) ◦ (cid:15) A , π ◦ ( F (cid:111) H )) ⊕ ((1 − π ) M (cid:111) H, , (1 − π ) ◦ ( F (cid:111) H )) . The second summand is degenerate. Hence the image of ((
M, ρ ) , φ, F ) under (4.24) is alsorepresented by ( π ( M (cid:111) H ) , ( φ (cid:111) H ) ◦ (cid:15) A , π ◦ ( F (cid:111) H )) . The map ψ : M → M (cid:111) H , m (cid:55)→ | H | (cid:88) h ∈ H ( m, h )induces a C -linear isomorphism of M with π ( M (cid:111) H ). For a in A and m in M we have ψ ( φ ( a ) m ) = ( φ (cid:111) H )( (cid:15) A ( a )) ψ ( m ) . Furthermore, ψ ( F ( m )) = π (( F (cid:111) H )( ψ ( m ))) . We equip M with the right- H -action by mh := ρ ( h − )( m ), then we get a covariant right-representation of B on M and hence a right-( B (cid:111) H )-module structure. We let M (cid:48) be M with this right ( B (cid:111) H )-module structure. The map ψ then becomes ( B (cid:111) H )-linear from M (cid:48) to π ( M (cid:111) H ). If we finally define the ( B (cid:111) H )-valued scalar product on M (cid:48) by (cid:104) m, m (cid:48) (cid:105) (cid:48) := 1 | H | (cid:88) h ∈ H ( (cid:104) ρ ( h )( m ) , m (cid:48) (cid:105) E , h − ) , then ψ becomes an isomorphism of right Hilbert ( B (cid:111) H )-modules from M (cid:48) to π ( M (cid:111) H ).We conclude that the image of (( M, ρ ) , φ, F ) under (4.24) is represented by the Kasparov( A, B (cid:111) H )-module ( M (cid:48) , φ, F ). 61or the inverse consider a class in KK ( A, B (cid:111) H ) represented by ( M (cid:48) , φ, F ). We consider M (cid:48) as a right Hilbert B -module M by restriction along B (cid:55)→ B (cid:111) H , b (cid:55)→ ( b, t : B (cid:111) H → B , (cid:88) h ∈ H ( b h , h ) (cid:55)→ b e and the B -valued scalar product on M by (cid:104) m, m (cid:48) (cid:105) := t ( (cid:104) m, m (cid:48) (cid:105) (cid:48) ) . We note that the right B -module M is essential so that we get a right H -action ( m, h ) (cid:55)→ mh on M . We then define ρ ( h )( m ) := mh . Then the inverse sends the class of ( M (cid:48) , φ, F ) tothe class of (( M, ρ ) , φ, F ).It is easy to see that these constructions are inverse to each other up to isomorphism. Sincethey preserve degenerate modules and are compatible with direct sums and homotopiesthey induce inverse to each other isomorphisms between Kasparov groups. This finishesthe verification that (4.24) is an isomorphism.Let B be in C ∗ Alg nu . Then we have a canonical homomorphism λ B : Res G ( B ) (cid:111) max G → B of C ∗ -algebras which corresponds to the covariant representation ( id B , triv) consistingof the identity of B and the trivial representation of G . The family λ = ( λ B ) B ∈ C ∗ Alg nu is a natural transformation λ : Res G ( − ) (cid:111) max G → id of endofunctors of C ∗ Alg nu . Thefollowing theorem is the spectrum-version of the dual Green-Julg theorem. It explicitlyidentifies the right-adjoint of − (cid:111) max G : KK G → KK whose existence was predicted byRemark 4.2.
Theorem 4.21.
The natural transformation λ is the counit of an adjunction − (cid:111) max G : KK G (cid:29) KK : Res G ( − ) . Proof.
We must show that the compositionGJ G : KK G ( − , Res G ( − )) − (cid:111) max G −−−−−→ KK( − (cid:111) max G, Res G ( − ) (cid:111) max G ) kk( λ ) ∗ −−−→ KK( − (cid:111) max G, − )(4.26)is an equivalence of functors from (KK G ) op × KK to Sp . In the first step we use the factthat the functors in the domain and target of (4.26) send filtered colimits in KK G to limits.Since KK G sep generates KK G under filtered colimits it suffices to show that the restrictionof the transformation to (KK G sep ) op × KK is an equivalence. But this restriction preservesfiltered colimits in KK. Hence it suffices to consider the restriction to (KK G sep ) op × KK sep . We thank J. Echterhoff for convincing us that the statement is true in this generality. π . It thus suffices to show that weget an isomorphism of groups KK G ( A, Res G ( B )) ∼ = −→ KK ( A (cid:111) max G, B ) (4.27)for every A in Fun ( BG, C ∗ Alg nusep ) and B in C ∗ Alg nusep . This is the classical dual Green–Julgtheorem (see [Bla98, Thm. 20.2.7] for the case B = C ).For completeness we sketch the argument. On level of Kasparov modules the map (4.27)sends the ( A, B )-module ((
H, ρ ) , φ, F ) to the ( A (cid:111) max G, B )-module ( H, ˜ φ, F ), where˜ φ : A (cid:111) max G → B ( H ) is the homomorphism canonically induced by the universal propertyof the maximal crossed product by the covariant representation ( φ, ρ ), where ρ is theunitary representation of G on H . Note that this module is essential in the sense that φ ( A (cid:111) max G )( H ) = H . For the inverse we use that we can actually represent every classin KK ( A (cid:111) max G, B ) by an essential module [Bla98, Prop. 18.3.6]. The inverse sendssuch a module ( H, ˜ φ, F ) to (( H, ρ ) , φ, F ), where φ is the restriction of ˜ φ via the canonicalembedding A → A (cid:111) max G which exists since G is discrete, and where ρ is the unitaryrepresentation on H induced by ˜ φ .This completes the proof of Theorem 1.22.
5. Locally finite K -homology Let X be in G Top prop+ and Y be a G -invariant closed subset of X . Then we have an exactsequence 0 → C ( X \ Y ) → C ( X ) → C ( Y ) → Fun ( BG, C ∗ Alg nu ). Recall the Definition 1.11 of the notion of split-closedness of Y . Proposition 5.1.
The closed invariant subset Y of X is split-closed in the following cases:1. G acts properly on an invariant neighbourhood of Y in X and Y is second countable.2. Y admits a G -invariant tubular neighbourhood.Proof. We start with the proof of Assertion 1. Let U be an invariant open neighbouhoodof Y in X . It suffices to construct an equivariant cpc split s : C ( Y ) → C ( U ). Composingwith the extension-by-zero map C ( U ) → C ( X ) we then get an equivariant cpc split forthe sequence in (5.1). 63ince Y is second countable, the C ∗ -algebra C ( Y ) is separable (see e.g. [Cho12]). Since it isalso nuclear, we can apply the Choi–Effros lifting theorem [CE76] (see also [Bla06, IV.3.2.5])in order to get a cpc split s : C ( Y ) → C ( U ) which is not necessarily equivariant. Sincehere we work with commutative algebras such a split is cpc if and only if it is contractiveand preserves positive functions.Since G acts properly on U we can choose a function χ in C ( U, [0 , G × U → U restricts to a proper map on G × supp ( χ ) and (cid:80) g ∈ G g − , ∗ χ ≡
1. Wenow define ¯ s : C ( Y ) → C ( U ) , ¯ s ( f ) := (cid:88) g ∈ G g − , ∗ χs ( g ∗ f ) . One verifies that this is now an equivariant cpc left inverse of the restriction map C ( U ) → C ( Y ).Under the assumption of Assertion 2 we have an invariant tubular neighbourhood U of Y ,an invariant retraction map r : U → Y , and an invariant radial function ρ : U → [0 , ∞ )such that r | ρ − ([0 , is proper. We choose a function χ : [0 , ∞ ) → [0 ,
1] such that χ (0) = 1and χ ( t ) = 0 for t ≥
1. Then we define s : C ( Y ) → C ( U ) by s ( f ) := ρ ∗ χ · r ∗ s . This is anequivariant cpc split of the restriction C ( U ) → C ( Y ). As above, composing with theextension-by-zero map C ( U ) → C ( X ) we get an equivariant cpc split for (5.1).We consider the functorkk G C ( − ) := kk G ◦ C ( − ) : ( G Top prop+ ) op → KK G . We furthermore let G Top prop2nd , + be the full subcategory of G Top prop+ consisting of secondcountable spaces and setkk G sep C ( − ) := kk G sep ◦ C ( − ) : ( G Top prop2nd , + ) op → KK G sep . These functors can be considered as the universal versions of the locally finite K -homologyfunctors from (1.3) and Definition 1.13. The following diagram commutes by definition( G Top prop2nd , + ) op kk G sep C (cid:47) (cid:47) incl (cid:15) (cid:15) KK G sep y G (cid:15) (cid:15) ( G Top prop+ ) op kk G C (cid:47) (cid:47) KK G (5.2)where y G is the functor from (1.6).In the following, we list the basic properties of these functors. The spaces in the statementbelong to G Top prop+ . Theorem 5.2. . The functor kk G C is homotopy invariant.2. If Y is an invariant split-closed subspace of X , then we have a natural fibre sequence kk G C ( X \ Y ) → kk G C ( X ) → kk G C ( Y ) .
3. We have kk G C ([0 , ∞ ) × X ) (cid:39) .4. If ( X n ) n ∈ N is a family in G Top prop2nd , + , then we have a canonical equivalence (cid:77) n ∈ N kk G sep C ( X n ) (cid:39) −→ kk G sep C (cid:0) (cid:71) n ∈ N X n (cid:1) .
5. The functor kk G C has a symmetric monoidal refinement kk G C ⊗ ? : G Top prop+ , ⊗ → KK G, ⊗ ? for ? in { min , max } .Proof. We start with Assertion 1. The functor C sends the projection [0 , × X → X tothe embdding C ( X ) → C ([0 , × X ) ∼ = C ([0 , ⊗ C ( X ) , which is an instance of (1.5). Since kk G is homotopy invariant, the assertion follows.We now show Assertion 2. If Y is a split-closed subspace of X , then we have a semisplitexact sequence 0 → C ( X \ Y ) → C ( X ) → C ( Y ) → , (5.3)where the first map is given by extension by zero, while the second map is the restrictionof functions from X to Y . We apply the semiexactness of kk G in order to get the desiredfibre sequence.For the Assertion 3 we consider the norm continuous path γ : [0 , ∞ ] → End C ∗ Alg nu ( C ([0 , ∞ ) × X ))given by γ t ( f )( x ) := f ( x + t ), where we set f ( ∞ ) := 0. It induces a homotopy C ([0 , ∞ ]) ⊗ C ([0 , ∞ ) × X ) → C ([0 , ∞ ) × X )from the identity to the zero map. We now use the homotopy invariance of kk G in orderto conclude that id kk G C ([0 , ∞ ) × X ) (cid:39) C (cid:0) (cid:71) n ∈ N X n (cid:1) ∼ = (cid:77) n ∈ N C ( X n ) .
65e then use that kk G sep preserves countable sums by Theorem 1.4.6.For Assertion 5 we use that the functor C : ( G Top prop+ , ⊗ ) op → Fun ( BG, C ∗ Alg nu ) ⊗ ? is symmetric monoidal, where the structure map is induced by C ( X ) ⊗ ? C ( X (cid:48) ) → C ( X × X (cid:48) ) , f ⊗ f (cid:48) (cid:55)→ (( x, x (cid:48) ) (cid:55)→ f ( x ) f ( x (cid:48) )) , see Remark 1.15 for the symmetric monoidal structure on G Top prop+ . This is true in bothcases ? = min and ? = max. We then use Proposition 1.20 stating that kk G is symmetricmonoidal. Remark 5.3.
In Assertion 5.2.2 we require that Y is split-closed. In fact, for an arbitraryinvariant closed subset Y of X we do not know whether the sequence (5.3) is semisplit.Furthermore, note that in Assertion 5.2.4 we must restrict to countable unions of secondcountable spaces since we only know that kk G sep preserves countable sums, while kk G is notexpected to have this property. Finally note that all the assertions stated for kk G C havean obvious version for kk G sep C . Example 5.4.
For every X in G Top prop2nd , + and n in N we have an equivalenceΣ n kk G C ( R n × X ) (cid:39) kk G C ( X ) . (5.4)In order to see this we argue by induction on n . The assertion is evidently true for n = 0.We now assume that n > G C ( R n × X ) (cid:47) (cid:47) (cid:15) (cid:15) kk G C ([0 , ∞ ) × R n − × X ) (cid:15) (cid:15) kk G C (( −∞ , × R n − × X ) (cid:47) (cid:47) kk G C ( R n − × X )where the maps are induced by the obvious inclusions of closed subspaces which aresplit-closed by Proposition 5.1.2. Since(( −∞ , × R n − × X ) \ ( R n − × X ) ∼ = ( R n × X ) \ ([0 , ∞ ) × R n − ) ∼ = ( −∞ , × X we conclude, using Theorem 5.2.2, that the vertical morphisms induce an equivalencebetween the cofibres of the horizontal maps. Therefore the square is a push-out square inKK G . We now use Theorem 5.2.3 in order to see that the lower-left and the upper-rightcorners are zero objects. The square therefore yields an equivalencekk G C ( R n − × X ) (cid:39) Σkk G C ( R n × X ) . By induction we now get (5.4). 66he following is Proposition 1.25 from the introduction. Recall the Definition 1.24.1 of G -proper objects in KK G . Proposition 5.5. If X is in G Top prop+ and proper homotopy equivalent to a finite G -CWcomplex with finite stabilizers, then kk G C ( X ) is G -proper.Proof. Since kk G C is homotopy invariant by Theorem 5.2.1 we can assume that X is a G -CW complex with finite stabilizers. We argue by a finite induction on the number of G -cells. Assume that X is obtained from X (cid:48) by attaching a G -cell of the form G/H × D n ,where H is a finite subgroup of G . Then X (cid:48) is a split-closed subspace of X by Proposition5.1.2 such that X \ X (cid:48) ∼ = G/H × R n . By Theorem 5.2.2 and (5.4) we have a fibre sequenceΣ − n kk G C ( G/H ) → kk G C ( X ) → kk G C ( X (cid:48) ) . Since Σ − n kk G C ( G/H ) is G -proper by definition and kk G C ( X (cid:48) ) is G -proper by inductionhypothesis we can conclude that kk G C ( X ) is G -proper, too.The following is Theorem 1.26 from the introduction. Theorem 5.6.
1. If P is ind- G -proper, then the functor KK G ( P, − ) : Fun ( BG, C ∗ Alg nu ) → Sp sends all exact sequences to fibre sequences.2. If P is G -proper, then the functor KK G ( P, − ) : Fun ( BG, C ∗ Alg nu ) → Sp preserves filtered colimits.Proof. We first show Assertion 1. The full subcategory of KK G of objects P such thatKK G ( P, − ) sends all exact sequences in Fun ( BG, C ∗ Alg nu ) to fibre sequences is localizing.In view of the Definition 1.24.2 of ind- G -properness it suffices to check that the functorKK G ( C ( G/H ) , − ) : Fun ( BG, C ∗ Alg nu ) → Sp sends exact sequences in Fun ( BG, C ∗ Alg nu ) to fibre sequences for all finite subgroups H of G .We will use that the usual K -theory functor for C ∗ -algebras (1.12) sends all exact sequencesto fibre sequences. By Corollary 1.23 we have the first equivalence inKK G ( C ( G/H ) , − ) (cid:39) KK( C , − (cid:111) H ) (1.12) = K C ∗ Alg ( − (cid:111) H ) . (5.5)67ince − (cid:111) H preserves exact sequences and K C ∗ Alg sends exact sequences in C ∗ Alg nu tofibre sequence we see that KK G ( C ( G/H ) , − ) sends exact sequences in Fun ( BG, C ∗ Alg nusep )to fibre sequences.The proof of Assertion 2 is completely analogous. The full subcategory of KK G of objects P such that KK G ( P, − ) : Fun ( BG, C ∗ Alg nu ) → Sp preserves filtered colimits is thick.Since the functors K C ∗ Alg and − (cid:111) H preserve filtered colimits we see that the categoryin question contains the algebras C ( G/H ) for all finite subgroups H of G . Consequentlyit contains all G -proper objects. Proof of Theorem 1.14.
The proof is based on the defining relation K G, lf A ( X ) (cid:39) KK G (kk G C ( X ) , A ) . (5.6)Assertion 1 is an immediate consequence of Proposition 1.19.Assertion 2 follows from Theorem 5.2.1.Assertion 3 follows by applying the exact functor KK G ( − , A ) to the fibre sequence inTheorem 5.2.2For Assertion 4 we consider the cases separately. If the sequence of algebras is semisplit,then the assertion immediately follows from the fact that kk G is semiexact by Theorem1.17.3. If it is just exact, then we use Theorem 5.6.1 and Proposition 5.5 instead. For thelast statement note that if X is second countable, then C ( X ) is separable, and hencekk G C ( X ) (cid:39) y G (kk G sep ( C ( X ))) is a compact object of KK G , see (5.2) and Remark 1.9.The first equivalence in Assertion 5 is an immediate consequence of Theorem 5.2.3. Thesecond equivalence uses Theorem 5.2.4 and the equivalence K G, lf A ( − ) (cid:39) map KK G sep (kk G sep C ( − ) , kk G sep ( A ))for separable A as functors on G Top prop2nd , + which follows from the fact that y G in (1.6) isfully faithful.In order to see Assertion 6 first note that for Y in G Top prop2nd , + and A in Fun ( BG, C ∗ Alg nusep )we have an equivalence K G, lf A ( Y ) (cid:39) map kk G sep (kk G sep ( C ( Y )) , kk G sep ( A )) . We use this expression in terms of the mapping spectrum in order to make it obvious thata colimit in the first argument can be pulled out as a limit. We can express the intersection Note that filtered colimits of G - C ∗ -algebras are calculated by forming the filtered colimit on the level ofunderlying vector spaces and then completing. Since H is finite, the functor − (cid:111) H commutes withboth operations.
68f the decreasing family of subspaces in categorical terms as a limit (cid:84) n ∈ N X n ∼ = lim n ∈ N X n which is interpreted in G Top prop+ . By Gelfand duality we get C (cid:0) (cid:92) n ∈ N X n (cid:1) ∼ = colim n ∈ N C ( X n ) , where the colimit is taken in Fun ( BG, C ∗ Alg nusep ). We now show that the diagram n (cid:55)→ C ( X n ) is admissible in the sense of [MN06, Def. 2.5]. According to the criterion [MN06,Lem. 2.7] it suffices to construct a family of equivariant cpc maps ( s n : C ( (cid:84) n ∈ N X n ) → C ( X n )) n ∈ N such that we have lim n →∞ ι n ◦ s n = id C ( (cid:84) n ∈ N X n ) in the norm topology, where( ι n : C ( X n ) → C ( (cid:84) n ∈ N X n )) n ∈ N is the family of restriction maps. By our assumption wecan choose an equivariant cpc left-inverse s : C ( (cid:84) n ∈ N X n ) → C ( X ) of the surjection ι .Then we can define the sought equivariant cpc maps s n as the composition of s with therestrictions C ( X ) → C ( X n ). Since ι n ◦ s n = id C ( (cid:84) n ∈ N X n ) for every n in N the family( s n ) n ∈ N has the required property. We finally use that kk G sep preserves filtered colimits ofadmissible diagrams by Theorem 1.4.7.The Assertion 7 follows from Corollary 1.23 together with (5.5).Finally, Assertion 8 follows from Theorem 5.2.5, Proposition 1.20, and the general factthat for a stable symmetric monoidal ∞ -category C the functor map : C op × C → Sp has anatural lax symmetric monoidal refinement.
6. Extension to C ∗ -categories In this section we consider the extension kk GC ∗ Cat (see Definition 1.27) of the functor kk G to C ∗ -categories. Basic references for C ∗ -categories are [GLR85], [Joa03], [Mit02], [Del12],[AV]. We will in particular use the language introduced in [Bun19], [Bun] which we recallin the following.We start with the category ∗ Cat nu C of possibly non-unital ∗ -categories which are C -vectorspace enriched categories C with an involution ∗ : C op → C which fixes objects and actsanti-linearly on morphisms. Morphisms in ∗ Cat nu C are functors which are compatible withthe involution and the enrichment. The category ∗ Cat nu C contains the full subcategory ∗ Alg nu of ∗ -algebras considered as categories with a single object. Using the uniquenessof the norm on a C ∗ -algebra and the automatic continuity of ∗ -homomorphisms between C ∗ -algebras we view the category C ∗ Alg nu of C ∗ -algebras as a full subcategory of ∗ Alg nu so that we can talk about functors from ∗ -categories to C ∗ -algebras.Given a ∗ -category C we define a maximal norm on morphisms by (cid:107) f (cid:107) max := sup ρ (cid:107) ρ ( f ) (cid:107) B ,where the supremum runs over all functors ρ to C ∗ -algebras B . This norm may be infinite.We call C a pre- C ∗ -category if all morphisms have a finite maximal norm. In this way weget the full subcategory pre C ∗ Cat nu of ∗ Cat nu C of pre- C ∗ -categories. Its intersection with69 Alg nu is the category pre C ∗ Alg nu of pre- C ∗ -algebras. A C ∗ -category is a pre- C ∗ -categoryin which all morphism spaces are complete with respect to the maximal norm. As explainedin [Bun19], [Bun] this definition is equivalent to other definitions in the literature. Again,we get the category C ∗ Alg nu of C ∗ -algebras by intersecting C ∗ Cat nu with ∗ Alg nu .There is also a unital version of all the above. The corresponding categories will be denotedin the same way but without the superscript ’nu’.The categories explained above are connected by adjunctionscompl : pre C ∗ Cat nu (cid:29) C ∗ Cat nu : incl and incl : pre C ∗ Cat nu (cid:29) ∗ Cat nu C : Bd ∞ (6.1)constructed in [Bun], where compl is the completion functor and the right adjoint Bd ∞ isthe bounded morphisms functor (since we only need the existence of Bd ∞ we refrain fromexplaining it more precisely). These adjunctions restrict correspondingly to adjunctionscompl : pre C ∗ Alg nu (cid:29) C ∗ Alg nu : incl and incl : pre C ∗ Alg nu (cid:29) ∗ Alg nu : Bd ∞ . (6.2)Using that ∗ Cat nu C is complete and cocomplete we can conclude formally from the existenceof the adjunctions (which are localizations or colocalizations, respectively) that all categoriesintroduced above are complete and cocomplete, as well. We refer to [Bun, Thm. 4.1] fordetails.While we define the extension kk GC ∗ Cat using the left adjoint functor A f from the adjunction(1.23) the verification of most of the properties of kk GC ∗ Cat uses another C ∗ -algebra A ( C )associated to a C ∗ -category. The problem with A ( − ) is that it is only functorial forfunctors between C ∗ -categories which are injective on objects so that A ( − ) can not directlybe used to define kk GC ∗ Cat . Construction 6.1.
Let C be in C ∗ Cat nu . Following [Joa03] we can form the object A alg ( C ) := (cid:77) C,C (cid:48) ∈ C Hom C ( C, C (cid:48) ) (6.3)of ∗ Alg nu with the obvious matrix multiplication and involution. One checks that A alg ( C )actually belongs to pre C ∗ Alg nu so that we define the object of C ∗ Alg nu A ( C ) := compl( A alg ( C )) (6.4)by applying the completion functor from (6.2).We have a canonical isometric functor C → A ( C ) which sends a morphism f : C → C (cid:48) in C to the corresponding one-entry matrix [ f C (cid:48) ,C ]. The construction of A ( C ) from C isfunctorial for functors in C ∗ Cat nu which are injective on objects. As further shown in[Joa03] the functor C → A ( C ) is initial for functors ρ : C → B with B a C ∗ -algebra withthe property that ρ ( f ) ρ ( f (cid:48) ) = (cid:40) ρ ( f ◦ f (cid:48) ) if the composition is defined0 else (6.5)70or any morphisms f, f (cid:48) in C .Let C → D be a functor in C ∗ Cat nu which is injective on objects and an isometricinclusion. Lemma 6.2.
The induced map A ( C ) → A ( D ) is an isometric inclusion.Proof. Let C (cid:48) be a full subcategory of C with finitely many objects, and let D (cid:48) be the fullsubcategory of D on the image of the objects of C (cid:48) . As said above, for every object C in C the map End C ( C ) → A ( C ) is an isometry, and similarly for objects of D . This easilyimplies that the upper horizontal and the vertical maps in A alg ( C (cid:48) ) (cid:47) (cid:47) (cid:15) (cid:15) A alg ( D (cid:48) ) (cid:15) (cid:15) A ( C ) (cid:47) (cid:47) A ( D )are isometries. Since A alg ( C ) is the union of the subalgebras of the form A alg ( C (cid:48) ) we con-clude that the maximal norm on this ∗ -algebra is the norm induced from the representationon A ( D ). This implies the assertion.Later we have to know that A ( − ) preserves certain filtered colimits. Let C : I → C ∗ Cat nu be a filtered diagram whose structure maps are injective on objects. Lemma 6.3.
The canonical map colim I A ( C ) → A ( colim I C ) is an isomorphism.Proof. As a formal consequence of the adjunctions in (6.1) we have the following formulafor colimits in C ∗ Cat nu : colim I C ∼ = compl( colim I ∗ Cat nu C C ) . (6.6)Here colim I ∗ Cat nu C C stands for the colimit interpreted in the category ∗ Cat nu C . The latterhappens to belong to pre C ∗ Cat nu so that we can apply the completion functor compl.We claim that for any D in pre C ∗ Cat nu there is canonical isomorphism A (compl( D )) ∼ = compl( A alg ( D )) . (6.7)71n order to show the claim we form the square D (1) (cid:47) (cid:47) (2) (cid:15) (cid:15) compl( D ) (3) (cid:15) (cid:15) A alg ( D ) (cid:40) (cid:40) (4) (cid:15) (cid:15) compl( A alg ( D )) (cid:47) (cid:47) A (compl( D ))The maps (1) and (4) are the canonical completion maps, the maps (3) and (2) are thecanonical morphisms from the categories to the corresponding algebras. The dotted arrowis induced from the universal property of (2) applied to the composition (3) ◦ (1). Finally,the dashed map comes from the universal property of (4) applied to the dotted arrow.This dashed arrow induces the desired isomorphism: In order to construct an inverse weconsider the diagram D (1) (cid:47) (cid:47) (2) (cid:15) (cid:15) compl( D ) (cid:124) (cid:124) (3) (cid:15) (cid:15) A alg ( D ) (4) (cid:15) (cid:15) compl( A alg ( D )) A (compl( D )) (cid:111) (cid:111) We get the dotted arrow from the universal property of (1) applied to (4) ◦ (2), and thenthe dashed arrow from the universal property of (3) applied to the dotted arrow. It isstraightforward to check that the dashed arrows in the two diagrams are inverse to eachother. This finishes the proof of the isomorphism (6.7).Since taking objects is a left adjoint [Bun, Lem. 2.4] and therefore commutes with colimitswe have a bijection Ob( colim I ∗ Cat nu C C ) ∼ = colim I Set
Ob( C ) . Furthermore, if
C, C (cid:48) are objects colim ∗ Cat nu C I C , then we can find i in I and objects ˜ C, ˜ C (cid:48) in C ( i ) such that ι i ( ˜ C ) = C and ι i ( ˜ C (cid:48) ) = C (cid:48) , where ι i : C ( i ) → colim I ∗ Cat nu C C is thecanonical functor. Then Hom colim ∗ Cat nu C I C ( C, C (cid:48) ) ∼ = colim ( κ : i → i (cid:48) ) ∈ I i/ Hom C ( i (cid:48) ) ( C ( κ )( ˜ C ) , C ( κ )( ˜ C (cid:48) )) . From this description and the formula (6.3) we easily conclude that colim I ∗ Alg nu A alg ( C ) ∼ = → A alg ( colim I ∗ Cat nu C C ) (6.8)72s an isomorphism. We get the isomorphisms colim I A ( C ) (6.4) ∼ = colim I compl( A alg ( C )) ! ∼ = compl( colim I ∗ Alg nu A alg ( C )) (6.8) ∼ = compl( A alg ( colim I ∗ Cat nu C C )) (6.7) ∼ = A (compl( colim I ∗ Cat nu C C )) (6.6) ∼ = A ( colim C C ) , where for the marked isomorphism we use that compl and the inclusion incl : pre C ∗ Alg nu → ∗ Alg nu are left-adjoints (see (6.2)) and therefore commute with all colimits. The colimitswithout superscripts are interpreted in C ∗ -algebras or C ∗ -categories, respectively.If C is in Fun ( BG, C ∗ Cat nu ), then we can form A ( C ) in Fun ( BG, C ∗ Alg nu ) where the G -action on A ( C ) is induced by functoriality. By the universal property of A f in (1.23)we have a canonical morphism in Fun ( BG, C ∗ Alg nu ) α C : A f ( C ) → A ( C ) , (6.9)natural for morphisms in Fun ( BG, C ∗ Cat nu ) which are injective on objects, such that C (cid:34) (cid:34) (cid:124) (cid:124) A f ( C ) α C (cid:47) (cid:47) A ( C )commutes. Here the left-down arrow is the unit of the adjunction (1.23).The following proposition is the main technical result which makes all other argumentsfurther below work. Proposition 6.4.
For every C in Fun ( BG, C ∗ Cat nu ) the morphism kk G ( α C ) : kk G ( A f ( C )) → kk G ( A ( C )) is an equivalence.Proof. We first assume that C has a countable set of non-zero objects (i.e., objects withnon-trivial endomorphism space), and that all morphism spaces of C are separable. In thiscase we repeat now the proof of [Joa03, Prop. 3.8]. We consider the separable G -Hilbertspace H := L ( { C } ∪ Ob (cid:54) =0 ( C )), where Ob (cid:54) =0 ( C ) is the set of non-zero objects in C andthe G -action is induced by the action of G on the set of objects of C . The additional G -fixed point { C } induces an embedding η : C → K ( H ). The argument in the citedreference provides a morphism β : A ( C ) → A f ( C ) ⊗ K ( H ). It furthermore shows thatthe composition β ◦ α (we omit the subscript C for better readability) is homotopic to id A f ( C ) ⊗ η , and the composition ( α ⊗ id K ( H ) ) ◦ β is homotopic to id A ( C ) ⊗ η . Using K G -stability and homotopy invariance of kk G we see that kk G ( β ) ◦ kk G ( α ) (cid:39) kk G ( id A f ( C ) ⊗ η )73s an equivalence, and that kk G ( β ) ◦ kk G ( α ) (cid:39) kk G ( β ) ◦ kk G ( α ⊗ id K ( H ) ) (cid:39) kk G ( id A ( C ) ⊗ η )is an equivalence, too. Consequently, kk G ( α ) is an equivalence.In the general case we use that C ∼ = colim C (cid:48) C (cid:48) , where the colimits run over the filteredposet of G -invariant subcategories C (cid:48) of C with countable sets of non-zero objects andseparable morphism spaces (recall that our standing assumption is that G is countable).Since A f is a left-adjoint it preserves filtered colimits in Fun ( BG, C ∗ Cat nu ). The functor A preserves the colimit above by Lemma 6.3. These two facts explain the lower verticalequivalences in the commutative diagram colim C (cid:48) kk G ( A f ( C (cid:48) )) colim C (cid:48) kk G ( α C (cid:48) ) (cid:39) (cid:47) (cid:47) (cid:15) (cid:15) colim C (cid:48) kk G ( A ( C (cid:48) )) (cid:15) (cid:15) kk G ( colim C (cid:48) A f ( C (cid:48) ) (cid:39) (cid:15) (cid:15) kk G ( colim C (cid:48) α C (cid:48) ) (cid:47) (cid:47) kk G ( colim C (cid:48) A ( C (cid:48) )) (cid:39) (cid:15) (cid:15) kk G ( A f ( C )) kk G ( α C ) (cid:47) (cid:47) kk G ( A ( C )) (6.10)In order to conclude that kk G ( α C ) is an equivalence it remains to show that the uppervertical morphisms are equivalences. To this end want to use that kk G is s-finitary.Therefore we must show that the sets of separable subalgebras A f ( C (cid:48) ) of A f ( C ) or A ( C (cid:48) )of A ( C ) are cofinal in the poset of all G -invariant separable subalgebras of A f ( C ) or A ( C ),respectively, where C (cid:48) runs over the G -invariant subcategories of C with countable sets ofnon-zero objects and separable morphism spaces.We first consider the algebra A ( C ). Let B be a G -invariant separable subalgebra of A ( C ).Let S be a countable dense subset of B . For every s in S we choose a sequence ( s n ) n ∈ N in A alg ( C ) such that lim n →∞ s n = s . The union of the G -orbits of the matrix elements of s n for all n and s in S together generate a G -invariant subcategory of C (cid:48) with countablymany non-zero objects and separable morphism spaces. We clearly have B ⊆ A ( C (cid:48) ).We now consider the case of A f . We use that A f ( C ) the constructed as the closure of thefree algebra A f, alg ( C ) generated by the morphisms of C subject to natural relations [Joa03,Def. 3.7]. Let B be a separable subalgebra of A f ( C ). Let S be a countable dense subsetof B . For every s in S we choose a sequence ( s n ) n ∈ N in A alg ( C ) such that lim n →∞ s n = s .We write s n as a finite linear combination of finite products of morphisms from C . Thisfinite set of morphisms will be called the set of components of s n (it is irrelevant that thisdefinition involves choices). We let C (cid:48) B be the G -invariant subcategory of C generatedby the union of G -orbits of the sets of components for all n in N and s in S . Then again C (cid:48) has at most countably many non-zero objects and separable morphism spaces, andwe have B ⊆ A f ( C (cid:48) ). This finishes the verification that the upper vertical morphisms in(6.10) are equivalences. We finally conclude that kk G ( α C ) is an equivalence.The following corollary of the proof of Proposition 6.4 settles Theorem 1.29.1. RecallDefinition 1.28 for the notion of an s -finitary functor.74 orollary 6.5. The functor kk GC ∗ Cat is s-finitary.Proof.
This is precisely the fact shown in the proof of Proposition 6.4 above that for every C in Fun ( BG, C ∗ Cat nu ) the left vertical composition in (6.10) is an equivalence.The notion of a unitary equivalence between C ∗ -categories is naturally defined in theunital, non-equivariant situation and extended to the equivariant case as follows. Let φ : C → D be a morphism in Fun ( BG, C ∗ Cat ). Definition 6.6. φ is a unitary equivalence if the underlying functor in C ∗ Cat is a unitaryequivalence.
Note that a unitary equivalence admits a weakly invariant (but in general not invariant)inverse up to a natural unitary isomorphism between weakly invariant functors, see [Bun,Def. 7.10 & Rem. 7.13] for further details.The following proposition verifies Theorem 1.29.2. Let φ : C → D be a morphism in Fun ( BG, C ∗ Cat ). Proposition 6.7. If φ : C → D is a unitary equivalence, then kk GC ∗ Cat ( φ ) : kk GC ∗ Cat ( C ) → kk GC ∗ Cat ( D ) is an equivalence.Proof. We first assume that C and D have countably many objects, that all morphismspaces in C and D are separable, and that φ is injective on objects. Then we have acommuting diagram in Fun ( BG, C ∗ Alg nusep ) A f ( C ) A f ( φ ) (cid:47) (cid:47) (cid:15) (cid:15) A f ( D ) (cid:15) (cid:15) A ( C ) A ( φ ) (cid:47) (cid:47) A ( D )Applying kk G we get a commuting squarekk GC ∗ Cat ( C ) kk GC ∗ Cat ( φ ) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) kk GC ∗ Cat ( D ) (cid:39) (cid:15) (cid:15) kk G ( A ( C )) kk G ( A ( φ )) (cid:47) (cid:47) kk G ( A ( D ))in KK G , where the vertical equivalences are justified by Proposition 6.4. It thereforesuffices to show that kk G ( A ( φ )) is an equivalence. To this end we will show that A ( φ ) is aKK G -equivalence. 75he equivalence φ gives rise to a Morita ( A ( C ) , A ( D ))-bimodule M constructed as theclosure of the algebraic sum (cid:77) C ∈ C ,D ∈ D Hom D ( D, φ ( C ))[BEa, Sec. 16]. The equivariant Kasparov module ( M, ρ,
0) represents an invertible elementof KK G ( A ( C ) , A ( D )), where ρ is the left-action of A ( C ) on M [Kas88, Thm. 2.18]. Wenow note that [ A ( φ )] in KK G ( A ( C ) , A ( D )) is induced by the equivariant Kasparov module( A ( D ) , A ( φ ) , A ( D ) ∼ = M ⊕ (cid:77) D (cid:48) ∈ D \ im ( φ ) ,D ∈ D Hom D ( D, D (cid:48) )such that A ( φ ) = ρ ⊕
0. It follows that [
M, ρ,
0] = [ A ( D ) , ρ ⊕ , ⊕
0] = [ A ( φ )]. Hence[ A ( φ )] is an invertible element in KK G ( A ( C ) , A ( D )). We conclude that A ( φ ) is a KK G -equivalence.We now drop the assumption that C and D have countably many objects and that themorphism spaces are separable. Let C (cid:48) and D (cid:48) be invariant subcategories of C and D with countably many non-zero objects and separable morphism spaces. Assume that ψ isa weakly invariant inverse of φ up to unitary natural transformation of weakly invariantfunctors. Then we can inductively enlarge C (cid:48) and D (cid:48) preserving the smallness properties inorder to obtain invariant subcategories C (cid:48)(cid:48) and D (cid:48)(cid:48) such that φ | C (cid:48)(cid:48) : C (cid:48)(cid:48) → D (cid:48)(cid:48) is a unitaryequivalence. The induction proceeds as follows: We set C (cid:48) := C (cid:48) and D (cid:48) := D (cid:48) . Assumethat C (cid:48) n and D (cid:48) n have been constructed.1. We enlarge C (cid:48) n to C (cid:48) n +1 such that Ob( C (cid:48) n +1 ) := Ob( C (cid:48) n ) ∪ Gψ (Ob( D (cid:48) n )) and themorphisms in C (cid:48) n +1 are generated by the morphisms in C (cid:48) n and the G -orbits ofthe morphisms in ψ ( D (cid:48) n ). Then C (cid:48) n +1 is again a G -invariant subcategory of C with countably many non-zero objects and separable morphism spaces such that C (cid:48) n ⊆ C (cid:48) n +1 .2. We now enlarge D (cid:48) n to D (cid:48) n +1 such that Ob( D (cid:48) n +1 ) := φ (Ob( C (cid:48) n +1 )) ∪ Ob( D (cid:48) n ). Themorphisms of D (cid:48) n +1 are generated by the morphisms of D (cid:48) n and the morphisms in φ ( C (cid:48) n +1 ). Then D (cid:48) n +1 is again a G -invariant subcategory of D with countably manynon-zero objects and separable morphism spaces such that D (cid:48) n ⊆ D (cid:48) n +1 .The induction yields increasing sequences of subcategories ( C (cid:48) n ) n ∈ N and ( D (cid:48) n ) n ∈ N . Theinvariant subcategories C (cid:48)(cid:48) := (cid:91) n ∈ N C (cid:48) n and D (cid:48)(cid:48) := (cid:91) n ∈ N D (cid:48) n have the required properties since ψ | D (cid:48)(cid:48) : D (cid:48)(cid:48) → C (cid:48)(cid:48) is a weakly invariant inverse equivalenceto φ | C (cid:48)(cid:48) . We conclude by the first step thatkk GC ∗ Cat ( φ | C (cid:48)(cid:48) ) : kk GC ∗ Cat ( C (cid:48)(cid:48) ) → kk GC ∗ Cat ( D (cid:48)(cid:48) )76s an equivalence. Using this construction in a cofinality consideration we obtain themiddle equivalence in the factorizationkk GC ∗ Cat ( C ) (cid:39) colim C (cid:48) kk GC ∗ Cat ( C (cid:48) ) (cid:39) colim D (cid:48) kk GC ∗ Cat ( D (cid:48) ) (cid:39) kk GC ∗ Cat ( D )of kk GC ∗ Cat ( φ ). The outer equivalences follow from the fact that kk GC ∗ Cat is s-finitary, seethe Corollary 6.5. This finishes the proof that kk GC ∗ Cat ( φ ) is an equivalence provided that φ is injective on objects.We finally drop the assumption that φ is injective on objects. Let φ : C → D be a unitaryequivalence. Then we form E in Fun ( BG, C ∗ Cat ) 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 equivalence j ◦ p ∼ = id E . We conclude that all morphisms i, j, p are unitaryequivalences. Since j is injective on objects, kk GC ∗ Cat ( j ) is an equivalence. From kk GC ∗ Cat ( p ) ◦ kk GC ∗ Cat ( j ) (cid:39) kk GC ∗ Cat ( id D ) we conclude that kk GC ∗ Cat ( p ) is an equivalence. Since i isinjective on objects, also kk G ( i ) is an equivalence. Hence kk GC ∗ Cat ( φ ) (cid:39) kk GC ∗ Cat ( p ) ◦ kk GC ∗ Cat ( i ) is an equivalence. Remark 6.8.
In [BEa, Sec. 15] we also introduce the notion of a relative unitary equiva-lence for ideals in unital C ∗ -categories or the notion of a partial isometric isomorphismbetween functors between possibly non-unital C ∗ -categories. We then showed that the K -theory functor K C ∗ Cat for C ∗ -categories sends relative unitary equivalences to equiva-lences or partial isometrically isomorphic functors to equivalent morphisms. The proofsdepend on the fact that K -theory for C ∗ -categories sends all exact sequences to fibresequences. Since kk GC ∗ Cat only has weaker exactness properties (stated in Proposition 6.14below) these proofs do not generalize to kk GC ∗ Cat in a straightforward manner.
Construction 6.9.
We let C ∗ Cat denote the subcategory of C ∗ Cat nu of unital C ∗ -categories and unital functors, and then define the ∞ -category C ∗ Cat ∞ as the Dwyer–Kan localization of C ∗ Cat at the unitary equivalences. As shown in [Del12], [Bun19] the77 -category C ∗ Cat ∞ is complete and cocomplete and modelled by a combinatorial modelcategory. By [Lur09, Sec. 4.2.4] or [Cis19, Prop. 7.9.2] the functor (cid:96) : Fun ( BG, C ∗ Cat ) → Fun ( BG, C ∗ Cat ∞ ) (6.11)exhibits Fun ( BG, C ∗ Cat ∞ ) as the Dwyer–Kan localization of Fun ( BG, C ∗ Cat ) againat the unitary equivalences. Since the functor kk GC ∗ Cat sends unitary equivalences toequivalences, we obtain a natural factorization
Fun ( BG, C ∗ Cat ) (cid:42) (cid:42) kk GC ∗ Cat (cid:47) (cid:47) KK G Fun ( BG, C ∗ Cat ∞ ) kk GC ∗ Cat , ∞ (cid:54) (cid:54) (6.12)by the universal property of the Dwyer–Kan localization.To prove Theorem 1.29.3, in the following two definitions we recall the notion of a weakMorita equivalence from [BEa]:Let D be in C ∗ Cat nu and S be a subset of the set D u of unital objects (i.e., objects whichadmit a unit endomorphism) in D . Definition 6.10 ([BEa, Def. 16.1]) . S is weakly generating if for every object D in D ,finite family ( A i ) i ∈ I of morphisms A i : D i → D in D , and any (cid:15) in (0 , ∞ ) there exists anisometry u : C → D such that (cid:107) A i − uu ∗ A i (cid:107) ≤ (cid:15) for all i in I and C is unitarily isomorphicto a finite orthogonal sum of objects in S . Let φ : C → D be a morphism in Fun ( BG, C ∗ Cat nu ). Definition 6.11 ([BEa, Def. 16.2]) . The functor φ is called a weak Morita equivalence if:1. C is unital,2. φ is fully faithfull,3. φ (Ob( C )) is weakly generating. Proposition 6.12.
The functor kk GC ∗ Cat sends weak Morita equivalences to equivalences.Proof.
We first assume that C and D have at most countably many non-zero objects andseparable morphism spaces. Using the same method as at the end of the proof of [BEa,78hm. 16.4] (see also the proof of Proposition 6.7 above) we can reduce to the case of weakMorita equivalences which are in addition injective on objects.Let φ : C → D be a weak Morita equivalence which is injective on objects. By Proposi-tion 6.4 and the fact that the image of y G in (1.6) generates KK G , we see that it sufficesto show that KK G ( A, A ( φ )) : KK G ( A, A ( C )) → KK G ( A, A ( D )) (6.13)is an equivalence for every A in Fun ( BG, C ∗ Alg nusep ). It furthermore suffices to show thatthis map induces an isomorphism on the level of homotopy groups.Our assumptions stated at the beginning of this proof imply that A ( C ) and A ( D ) belongto Fun ( BG, C ∗ Alg nusep ). It suffices to show that KK G ∗ ( A, A ( φ )) : KK G ∗ ( A, A ( C )) → KK G ∗ ( A, A ( D ))is an isomorphism. In the proof of [BEa, Thm. 16.3] we have constructed an equivariantMorita ( A ( C ) , A ( D ))-bimodule (the G -action is induced by naturality of the construction)which induces the map KK G ∗ ( A, A ( φ )) on the level of Kasparov modules. Since thereexists an inverse Morita ( A ( D ) , A ( C ))-bimodule it is now clear that KK G ∗ ( A, A ( φ )) is anisomorphism. We conclude that kk GC ∗ Cat ( φ ) : kk GC ∗ Cat ( C ) → kk GC ∗ Cat ( D ) is an equivalenceunder our additional assumptions on C and D .In order to extend to the general case we again use that the functor kk GC ∗ Cat is s-finitaryby Corollary 6.5.Let C (cid:48) and D (cid:48) be invariant subcategories of C and D , respectively, with at most countablymany non-zero objects and separable morphism spaces. We choose a countable densesubset M of the morphisms of D . For every non-zero D (cid:48) in D (cid:48) and finite family ( A i ) i with A i : D (cid:48) i → D (cid:48) in M and n in N we choose a finite family ( C j ) j ∈ J of objects in C and anisometry u : (cid:76) j ∈ J C j → D (cid:48) such that (cid:107) A i − uu ∗ A i (cid:107) ≤ n for all i in I . This is possiblesince φ is a weak Morita equivalence.We define the set objects of the invariant subcategory C (cid:48)(cid:48) of C as the set of objects of C (cid:48) together with all G -orbits of the objects C j appearing in these families above. We then let D (cid:48)(cid:48) be the invariant, finitely additive closure in D generated by D (cid:48) , φ ( C (cid:48) ), the identitiesof the objects of φ ( C (cid:48)(cid:48) ), and all the morphisms u and the structure maps of the sumsappearing above. Using that φ is fully faithful we then define the morphisms of C (cid:48)(cid:48) suchthat φ restricts to a fully faithful functor φ | C (cid:48)(cid:48) : C (cid:48)(cid:48) → D (cid:48)(cid:48) . Then φ : C (cid:48)(cid:48) → D (cid:48)(cid:48) is a weakMorita equivalence. The categories C (cid:48)(cid:48) and D (cid:48)(cid:48) have at most countably many non-zeroobjects and separable morphism spaces.Using these observations in a cofinality argument we get the middle equivalence in thefactorization of kk GC ∗ Cat ( φ ) askk GC ∗ Cat ( C ) (cid:39) colim C (cid:48) kk GC ∗ Cat ( C (cid:48) )) (cid:39) −→ colim D (cid:48) kk GC ∗ Cat ( D (cid:48) ) (cid:39) kk GC ∗ Cat ( D ) , C (cid:48) or D (cid:48) , respectively, having at most countablymany non-zero objects and separable morphism spaces.The following proposition shows Theorem 1.29.4. Proposition 6.13.
We have an equivalence kk C ∗ Cat ( − (cid:111) ? G ) (cid:39) ( − (cid:111) ? G ) ◦ kk GC ∗ Cat of functors from
Fun ( BG, C ∗ Cat nu ) → KK for ? ∈ { r, max } .Proof. We have the following chain of equivalenceskk C ∗ Cat ( − (cid:111) ? G ) Def. 1 . = kk( A f ( − (cid:111) ? G )) Prop. 6 . (cid:39) kk( A ( − (cid:111) ? G )) ! (cid:39) kk( A ( − ) (cid:111) ? G ) (4.12) (cid:39) ( − (cid:111) ? G ) ◦ kk G ( A ( − )) Prop. 6 . (cid:39) ( − (cid:111) ? G ) ◦ kk G ( A f ( − )) Def. 1 . = ( − (cid:111) ? G ) ◦ kk GC ∗ Cat ( − ) , where for ! we use the fact shown in [Bun, Thm. 6.9] that A ( − ) preserves the maximalcrossed product, resp. [BEa, Thm. 10.13] that A ( − ) preserves the reduced crossed product.An exact sequence 0 → A → B → C → Fun ( BG, C ∗ Cat nu ) is a sequence of functorswhich induce bijections between the sets of objects and exact sequences0 → Hom A ( C, C (cid:48) ) → Hom B ( C, C (cid:48) ) → Hom C ( C, C (cid:48) ) → C, C (cid:48) in C (which will be considered also as objects of A and B in thenatural way). This is equivalent to the definition of an exact sequence of G - C ∗ -categoriesgiven in [Bun, Def. 8.3].The following proposition shows Theorem 1.29.5 and 1.29.6. Recall Definition 1.24 of G -properness and ind- G -properness. Proposition 6.14.
1. If P in KK G is ind- G -proper, then the functor KK G ( P, kk GC ∗ Cat ( − )) sends all exactsequences in Fun ( BG, C ∗ Cat nu ) to fibre sequences. . If P in kk G is G -proper, then KK G ( P, kk GC ∗ Cat ( − )) preserves filtered colimits.Proof. We start with Assertion 1. Let0 → A → B → C → Fun ( BG, C ∗ Cat nu ). Then by [Bun, Prop. 8.8] we have an exactsequence 0 → A ( A ) → A ( B ) → A ( C ) → Fun ( BG, C ∗ Alg nu ). By Theorem 1.26.1 and the assumption on P we then get the fibresequence KK G ( P, A ( A )) → KK G ( P, A ( B )) → KK G ( P, A ( C )) . Finally, we get the desired fibre sequenceKK G ( P, kk GC ∗ Cat ( A )) → KK G ( P, kk GC ∗ Cat ( B )) → KK G ( P, kk GC ∗ Cat ( C )))from Proposition 6.4.In order to show Assertion 2 we note that A f preserves all colimits since it is a left adjoint.The assertion now immediately follows from Theorem 1.26.2.In the following we need the notion of a weakly invariant functors and of natural transfor-mations between them from [Bun, Def. 7.10]. Furthermore we will use that the orthogonalsum of two weakly invariant functors with unital target is again weakly invariant in acanonical way, see [BEa, Ex. 9.9]. Let C be Fun ( BG, C ∗ Cat ). The following definitiongeneralizes [BEa, Def. 9.3] from the non-equivariant to the equivariant case. Recall that C is called additive if the underlying C ∗ -category obtained by forgetting the G -action admitsorthogonal sums for all finite families of objects [BEa, Def. 3.5]. Definition 6.15. C is called flasque if it is additive and admits a weakly invariantendomorphism S : C → C such that we have a natural unitary isomorphism of weaklyinvariant functors S ∼ = S ⊕ id C . The following proposition shows the remaining Assertion 7 of Theorem 1.29.
Proposition 6.16. If C in Fun ( BG, C ∗ Cat ) is flasque, then KK G ( P, kk GC ∗ Cat ( C )) (cid:39) for all ind- G -proper P in KK G .Proof. Assume that C in Fun ( BG, C ∗ Cat ) is flasque. The full subcategory of objects P in KK G such that KK G ( P, kk GC ∗ Cat ( C )) (cid:39) G ( C ( G/H ) , kk GC ∗ Cat ( C )) (cid:39) of G . In this case by Definition 1.27, Proposition 6.4 and Corollary 1.23 we have thefollowing equivalences (we omit Res GH to simplify the notation):KK G ( C ( G/H ) , kk GC ∗ Cat ( C )) (cid:39) KK G ( C ( G/H ) , A f ( C )) (cid:39) KK G ( C ( G/H ) , A ( C )) (cid:39) KK( C , A ( C ) (cid:111) H ) . We now use that A ( − ) commutes with crossed products [Bun, Thm. 6.9] and againProposition 6.4 (for the trivial group) and the definition of K C ∗ Cat := KK ( C , A f ( − )) inorder to get an equivalence KK ( C , A ( C ) (cid:111) H ) (cid:39) KK ( C , A ( C (cid:111) H )) (cid:39) KK ( C , A f ( C (cid:111) H )) (cid:39) K C ∗ Cat ( C (cid:111) H ) . We claim that C (cid:111) H is again flasque. Let S : C → C be the weakly invariant functorimplementing the flasqueness of C such that we have a natural unitary S ⊕ id C ∼ = S ofweakly invariant functors. By [Bun, Prop. 7.12] the crossed product is functorial withrespect to weakly invariant functors and unitary transformations between them. Hence weget a unitary isomorphism S (cid:111) H ⊕ id C (cid:111) H ∼ = ( S ⊕ id C ) (cid:111) H ∼ = S (cid:111) H .
Hence S (cid:111) H implements the flasqueness of C (cid:111) H . By [BEa, Prop. 11.12 & Thm. 12.4] thefunctor K C ∗ Cat annihilates flasques so that K C ∗ Cat ( C (cid:111) H ) (cid:39)
0. By going back throughthe equivalences above we conclude that KK G ( C ( G/H ) , kk GC ∗ Cat ( C )) (cid:39) K -theory of stable multiplier algebras is trivial[Bla98, Sec. 12]. As an application of the techniques developed so far we will provide ageneralization of this fact to the equivariant situation, see Corollary 6.18 below.Let H := (cid:96) ⊗ L ( G ) be the standard G -Hilbert space. Let A be in Fun ( BG, C ∗ Alg nu ).Then we consider the stable multiplier algebra M ( A ⊗ K ( H )) of A as a G - C ∗ -categorywith a single object. Lemma 6.17.
The category M ( A ⊗ K ( H )) is flasque.Proof. We will show that M ( A ⊗ K ( H )) is countably additive. This will finish the proofsince countably additive C ∗ -categories are flasque.For any countable set I we can choose a pairwise orthogonal family of isometries ( u i ) i ∈ I , u i : (cid:96) → (cid:96) , such that (cid:80) i ∈ I u i u ∗ i = id (cid:96) in the weak operator topology. We have anisomorphism K ( H ) ∼ = K ( (cid:96) ) ⊗ K ( L ( G )) and define the multiplier e i in M ( A ⊗ K ( H )) suchthat e i ( a ⊗ b ⊗ c ) := a ⊗ u i b ⊗ c for every a in A , b in K ( (cid:96) ) and c in K ( L ( G )). Then ( e i ) i ∈ I is a mutually orthogonal family of isometries such that (cid:80) i ∈ I e i e ∗ i = 1 M ( A ⊗ K ( H )) , where thesum converges strictly. By [BEa, Prop. 5.10] the pair ( M ( A ⊗ K ( H )) , ( e i ) i ∈ I ) representsthe orthogonal sum of the family ( M ( A ⊗ K ( H ))) I in the C ∗ -category M ( A ⊗ K ( H )).The latter is therefore countably additive. 82ombining Lemma 6.17 with Proposition 6.16 immediately implies: Corollary 6.18.
We have KK G ( P, M ( A ⊗ K ( H ))) (cid:39) for all ind- G -proper P in KK G . If G is trivial and P = kk( C ), then this is, as already noted above, the well-known resultthat the K -theory of stable multiplier algebras is trivial. The classical proof is differentand shows that the unitary group of such an algebra is contractible.
7. Tensor products of C ∗ -categories The main goal of the present section is to prove Theorem 1.31 from the introduction statingthat kk GC ∗ Cat has symmetric monoidal refinements for the maximal and minimal tensorproducts on C ∗ -categories. In order to show this result we give an essentially completeaccount for the maximal and minimal tensor products on C ∗ Cat nu . As an easy consequenceof the definitions in Corollary 7.7 we obtain an op-lax symmetric monoidal refinementof kk GC ∗ Cat in both cases. So the main (and most complicated part) is the verification,stated as Proposition 7.8, that this structure is actually symmetric monoidal. The crucialtechnical result used in its proof is the Proposition 7.9 asserting the compatibility of A from (6.4) with the minimal and maximal tensor products of C ∗ -categories.Our starting point is the symmetric monoidal structure on ∗ Cat nu C given by the algebraictensor product. Definition 7.1.
For C and D in ∗ Cat nu C the algebraic tensor product is characterized bythe property that the morphism C × D → C ⊗ alg D in ∗ Cat nu (possibly non-unital categories with involution) is universal for morphisms from C × D to objects from ∗ Cat nu C which are bilinear on morphism spaces. Here is an description of the algebraic tensor product of C and D in ∗ Cat nu C :1. objects: We have Ob( C ⊗ alg D ) ∼ = Ob( C ) × Ob( D ).2. morphisms: For objects ( C, D ) and ( C (cid:48) , D (cid:48) ) in C ⊗ alg D we have Hom C ⊗ alg D (( C, D ) , ( C (cid:48) , D (cid:48) )) ∼ = Hom C ( C, C (cid:48) ) ⊗ Hom D ( D, D (cid:48) ) .
83. composition and involution: These structures are defined in the obvious manner.The maximal tensor product in C ∗ Cat or C ∗ Cat nu has a similar description by a universalproperty: Definition 7.2.
For C and D in C ∗ Cat nu the maximal tensor product is characterizedby the property that the morphism C × D → C ⊗ max D in ∗ Cat nu is universal for morphisms from C × D to objects from C ∗ Cat nu which arebilinear on morphism spaces. One must check that the maximal tensor product exists. In the unital case this has beenshown in [Del12, Prop. 3.12], but the proof given there explicitly uses identity morphismsand does not directly apply in the non-unital case. The first step in the verification is thefollowing lemma. Assume that C and D are in C ∗ Cat nu . Lemma 7.3.
The algebraic tensor product C ⊗ alg D is a pre- C ∗ -category.Proof. It suffices to check that f ⊗ g has a finite maximal norm for every pair of morphisms f in C and g in D . We will show that (cid:107) f ⊗ g (cid:107) max ≤ (cid:107) f (cid:107) C (cid:107) g (cid:107) D . (7.1)Let ρ : C ⊗ alg D → A be a functor into to a C ∗ -algebra (considered as a morphism in ∗ Cat nu C ). Then we will show that (cid:107) ρ ( f ⊗ g ) (cid:107) A ≤ (cid:107) f (cid:107) C (cid:107) g (cid:107) D . This fact is well-known forhomomorphisms from algebraic tensor products on C ∗ -algebras [Mur90, Cor. 6.3.6]. Usingthe C ∗ -equality for the norm on C ∗ -categories, the case of C ∗ -categories can be reducedto the case of C ∗ -algebras as follows. We have (cid:107) ρ ( f ⊗ g ) (cid:107) A = (cid:107) ρ ( f ∗ ⊗ g ∗ ) ρ ( f ⊗ g ) (cid:107) A = (cid:107) ρ ( f ∗ f ⊗ g ∗ g ) (cid:107) A ≤ (cid:107) f ∗ f (cid:107) C (cid:107) g ∗ g (cid:107) D = (cid:107) f (cid:107) C (cid:107) g (cid:107) D , where for the inequality we use that ρ induces a representation of the algebraic tensorproduct of C ∗ -algebras End C ( C ) ⊗ alg End D ( D ) to A . Since ρ is arbitrary the inequality(7.1) follows. Proposition 7.4.
The maximal tensor product ⊗ max on C ∗ Cat nu exists and equips thiscategory with a symmetric monoidal structure.Proof. In view of Lemma 7.3 the algebraic tensor product induces a symmetric monoidalfunctor C ∗ Cat nu → pre C ∗ Cat nu . Using the completion functor we define C ⊗ max D := compl( C ⊗ alg D ) .
84t remains to define the unit, associativity and symmetry constraints. Thereby only theassociativity is not completely straightforward. In order to construct it we consider thebold part of the commutative diagram( A ⊗ alg B ) ⊗ alg C ∼ = (cid:47) (cid:47) (cid:15) (cid:15) A ⊗ alg ( B ⊗ alg C ) (cid:15) (cid:15) ( A ⊗ max B ) ⊗ alg C (cid:42) (cid:42) (cid:15) (cid:15) A ⊗ alg ( B ⊗ max C ) (cid:15) (cid:15) ( A ⊗ max B ) ⊗ max C (cid:47) (cid:47) A ⊗ max ( B ⊗ max C )whose vertical morphisms are all given by the unit of the first adjunction in (6.1) andthe functoriality of the algebraic tensor product. The upper horizontal functor is theassociativity constraint of the algebraic tensor product. We obtain the dotted arrow fromthe universal property of the algebraic tensor product: To this end we must show that thebilinear functor ( A ⊗ alg B ) × alg C → A ⊗ max ( B ⊗ max C )induced by the right-down composition extends by continuity to a bilinear functor( A ⊗ max B ) × alg C → A ⊗ max ( B ⊗ max C ) . For a morphism φ in A ⊗ alg B and h in C we have by (7.1) that (cid:107) φ ⊗ h (cid:107) A ⊗ max ( B ⊗ max C ) ≤ (cid:107) φ (cid:107) max (cid:107) h (cid:107) C . This estimate implies that the bilinear functor extends as desired, and the existence of thedotted arrow follows.We finally get the dashed arrow from the universal property of the lower left vertical arrowapplied to the dotted arrow. In order to show that it is an isomorphism we construct aninverse by a similar argument starting from the inverse of the upper horizontal arrow.It is clear from the universal property of ⊗ max , or alternatively from its construction,that the inclusion functor incl : C ∗ Alg nu → C ∗ Cat nu has a canonical symmetric monoidalrefinement for the maximal tensor structures on the domain and the target.We now turn to the minimal tensor product on C ∗ Cat nu .The category Hilb of small Hilbert spaces is a commutative algebra in ∗ Cat nu C such thatthe structure morphism Hilb ⊗ alg Hilb → Hilb is induced by the universal property of ⊗ alg by the functor Hilb × Hilb → Hilb given asfollows: 85. objects: A pair (
H, H (cid:48) ) of Hilbert spaces is sent to H ⊗ H (cid:48) (tensor product in thesense of Hilbert spaces).2. morphisms: A pair of morphism ( f, g ) : ( H , H (cid:48) ) → ( H , H (cid:48) ) is sent to the morphism f ⊗ g : H ⊗ H (cid:48) → H ⊗ H (cid:48) .The unit of this algebra is the inclusion functor C → Hilb .Let C , D be in C ∗ Cat nu and c : C → Hilb and d : D → Hilb be functors. Then we candefine a functor c ⊗ d : C ⊗ alg D → Hilb ⊗ alg Hilb → Hilb . Definition 7.5.
The minimal tensor product C ⊗ min D is defined as the completion ofthe algebraic tensor product such that for every c, d as above we have a factorization C ⊗ alg D c ⊗ d (cid:47) (cid:47) (cid:39) (cid:39) Hilb . C ⊗ min D (cid:56) (cid:56) In other words, the minimal norm of a morphism φ in C ⊗ alg D is given by (cid:107) φ (cid:107) min := sup c,d (cid:107) ( c ⊗ d )( φ ) (cid:107) Hilb . (7.2) Proposition 7.6.
The minimal tensor product ⊗ min equips C ∗ Cat nu with a symmetricmonoidal structure.Proof. We must provide the unit, associativity, and symmetry constraints. As in the caseof the maximal tensor product only the associativity constraint is non-straightforward. Inorder to construct it we consider the bold part of the commutative diagram( A ⊗ alg B ) ⊗ alg C ∼ = (cid:47) (cid:47) (cid:15) (cid:15) A ⊗ alg ( B ⊗ alg C ) (cid:15) (cid:15) ( A ⊗ min B ) ⊗ alg C (cid:42) (cid:42) (cid:15) (cid:15) A ⊗ alg ( B ⊗ min C ) (cid:15) (cid:15) ( A ⊗ min B ) ⊗ min C (cid:47) (cid:47) A ⊗ min ( B ⊗ min C )where the vertical maps are given by the canonical maps from the algebraic tensor productsto the respective completions. 86s in the case of the maximal tensor product, in order to show the existence of the dottedarrow we must show that the bilinear functor( A ⊗ alg B ) × alg C → A ⊗ min ( B ⊗ min C )induced by the right-down composition extends by continuity to a bilinear functor( A ⊗ min B ) × alg C → A ⊗ min ( B ⊗ min C ) . Let a : A → Hilb , b : B → Hilb and c : C → Hilb be representations. Let φ be in( A ⊗ alg B ) and h be in C . Then we have the inequalities (cid:107) ( a ⊗ b ⊗ c )( φ ⊗ h ) (cid:107) Hilb ≤ (cid:107) ( a ⊗ b )( φ ) (cid:107) Hilb (cid:107) c ( h ) (cid:107) Hilb ≤ (cid:107) φ (cid:107) min (cid:107) h (cid:107) C . Since a, b, c are arbitrary we conclude that (cid:107) φ ⊗ h (cid:107) A ⊗ min ( B ⊗ min C ) ≤ (cid:107) φ (cid:107) min (cid:107) h (cid:107) C . Thisestimate implies that the bilinear functor extends as desired and that the dotted arrowexists.The first part of the estimate above shows that the dotted arrow further extends bycontinuity to the dashed arrow. An inverse of the dashed arrow can be constructed in asimilar manner starting from the inverse of the upper horizontal arrow.It is again clear from the universal property of ⊗ min , or alternatively from the constructionof the minimal norm in (7.2), that the inclusion functor incl : C ∗ Alg nu → C ∗ Cat nu has acanonical symmetric monoidal refinement for the minimal tensor structures on the domainand the target.Let us now collect some facts about the minimal tensor product which we will use atvarious places in the present section.If A is in C ∗ Alg nu , then a representation α : A → Hilb of A is the same datum as ahomomorphism α : A → B ( H ) for some Hilbert space H . If β : B → B ( H (cid:48) ) is a secondhomomorphism, then their tensor product in the sense of representations to Hilb is simplythe tensor product α ⊗ β : A ⊗ alg B → B ( H ) ⊗ B ( H (cid:48) ) ∼ = B ( H ⊗ H (cid:48) ) . It is known that if α and β are faithful representations, then (cid:107) x (cid:107) min = (cid:107) ( α ⊗ β )( x ) (cid:107) B ( H ⊗ H (cid:48) ) (7.3)for all x in A ⊗ alg B . Thus for C ∗ -algebras the supremum in (7.2) is realized by any pairof faithful representations. Corollary 7.7.
The functor kk GC ∗ Cat canonically refines to an op-lax symmetric monoidalfunctor kk G, ⊗ ? C ∗ Cat : C ∗ Cat nu , ⊗ ? → KK G, ⊗ ? for ? ∈ { min , max } . roof. As observed previously, the inclusion functor in (1.23) has a symmetric monoidalrefinement for the structures ⊗ ? . Hence its left-adjoint A f acquires a canonical op-laxsymmetric monoidal structure. Since kk G is symmetric monoidal by Proposition 1.20we conclude that the composition kk GC ∗ Cat = kk G ◦ A f has a canonical op-lax symmetricmonoidal structure.The following proposition finishes the verification of Theorem 1.31 from the introduction. Proposition 7.8.
For ? ∈ { min , max } the op-lax symmetric monoidal functor kk G, ⊗ ? C ∗ Cat : C ∗ Cat nu , ⊗ ? → KK G, ⊗ ? is symmetric monoidal. Let C , D be in C ∗ Cat nu . Then we have functors C → A ( C ) and D → A ( D ). We considerthe composition C × D → A ( C ) × A ( D ) → A ( C ) ⊗ ? A ( D )in ∗ Cat nu . This functor is bilinear and hence, by the universal property of the respectivetensor products, factorizes uniquely over the functor i in C × D (cid:15) (cid:15) (cid:40) (cid:40) C ⊗ ? D i (cid:47) (cid:47) A (cid:15) (cid:15) A ( C ) ⊗ ? A ( D ) .A ( C ⊗ ? D ) (cid:54) (cid:54) We now use the universal property of the functor A formulated in Construction 6.1 (theconditions are straightforward to check), that the functor i further factorizes over thedashed homomorphism as indicated. We will call this the canonical homomorphism inwhat follows. Proposition 7.9.
For all C , D in C ∗ Cat nu and ? in { min , max } the canonical homomor-phism A ( C ⊗ ? D ) → A ( C ) ⊗ ? ( D ) (7.4) is an isomorphism Remark 7.10.
One can use Proposition 7.9 in the case ? = max in order to show thatthe definition of the maximal tensor product of C ∗ -categories given in [AV, Sec. 3.1] isequivalent to the Definition 7.2 used in the present paper.88 roof of Prop. 7.8 assuming Prop. 7.9. Let C and D be in Fun ( BG, C ∗ Cat nu ). Thestructure map of the op-lax symmetric monoidal structure on A f is a homomorphism A f ( C ⊗ ? D ) → A f ( C ) ⊗ ? A f ( D ) . (7.5)We must show that the morphismkk G ( A f ( C ⊗ ? D )) kk G ((7.5)) −−−−−−→ kk G ( A f ( C ) ⊗ ? A f ( D )) (cid:39) kk G ( A f ( C )) ⊗ ? kk G ( A f ( D ))is an equivalence in KK G , where the second equivalence is the inverse of the structure mapof the symmetric monoidal structure of kk G . It is easy to see that we have the followingcommutative diagram A f ( C ⊗ ? D ) (7.5) (cid:47) (cid:47) α C ⊗ ? D (cid:15) (cid:15) A f ( C ) ⊗ ? A f ( D ) α C ⊗ ? α D (cid:15) (cid:15) A ( C ⊗ ? D ) ∼ =(7.4) (cid:47) (cid:47) A ( C ) ⊗ ? A ( D )where the vertical morphisms are induced by instances of (6.9), and the lower horizontalmap is an isomorphism by the assumed Proposition 7.9. We now apply kk G and getkk G ( A f ( C ⊗ ? D )) kk G ((7.5)) (cid:47) (cid:47) kk G ( α C ⊗ ? D ) (cid:15) (cid:15) kk G ( A f ( C ) ⊗ ? A f ( D )) kk G ( α C ⊗ ? α D ) (cid:15) (cid:15) kk G ( A ( C ⊗ ? D )) (cid:39) (cid:47) (cid:47) kk G ( A ( C ) ⊗ ? A ( D )) . Using Proposition 6.4 for the left vertical arrow and Proposition 1.20 for the equivalencekk G ( α C ⊗ ? α D ) (cid:39) kk G ( α C ) ⊗ ? kk G ( α D )in order to deal with the right vertical arrow, we conclude that the vertical arrows areequivalences. We conclude that kk G ((7.5)) is an equivalence.The following lemma is the first step of the proof of Proposition 7.9. Lemma 7.11. If C , D in C ∗ Cat nu have at most finitely many objects, then the canonicalhomomorphism A ( C ⊗ ? D ) → A ( C ) ⊗ ? A ( D ) is an isomorphism for ? in { min , max } .Proof. The assumption implies that A alg ( C ) → A ( C ) and A alg ( D ) → A ( D ) are isomor-phisms, where A alg is as in (6.3). Since the algebraic tensor product in ∗ Alg nu is formedon the level of underlying complex vector spaces and direct sums commute with tensorproducts we furthermore conclude that the algebraic analog A ( C ⊗ alg D ) → A ( C ) ⊗ alg A ( D )89f the canonical homomorphism is an isomorphism. We now consider the diagram A ( C ⊗ alg D ) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) A ( C ⊗ ? D ) A ( C ) ⊗ alg A ( D ) (cid:47) (cid:47) A ( C ) ⊗ ? A ( D ) (cid:79) (cid:79) If ? = max, then we obtain the dotted arrow from the universal property of the lowerhorizontal arrow applied to the up-right composition.In the case of ? = min we argue as follows. In order to show that the dotted arrowexists, by the universal property of ⊗ min on C ∗ Alg nu we must show that for every pairof representations c : A ( C ) → Hilb and d : A ( D ) → Hilb we have a factorization asindicated by the dashed arrow in the extended diagram C ⊗ alg D (cid:15) (cid:15) (cid:47) (cid:47) C ⊗ ? D c (cid:48) ⊗ d (cid:48) (cid:39) (cid:39) ! (cid:15) (cid:15) A ( C ⊗ alg D ) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) A ( C ⊗ ? D ) (cid:47) (cid:47) Hilb .A ( C ) ⊗ alg A ( D ) (cid:47) (cid:47) A ( C ) ⊗ ? A ( D ) c ⊗ d (cid:55) (cid:55) To this end we consider the dotted part of the diagram, where c (cid:48) : C → Hilb and d (cid:48) : D → Hilb are the restrictions of c and d along C → A ( C ) and D → A ( D ). The arrow c (cid:48) ⊗ d (cid:48) exists by the universal property of the minimal tensor product on C ∗ Cat nu . Weget the dashed arrow from the universal property of the arrow marked by !.In order to see in both cases of ? that the homomorphism A ( C ) ⊗ ? A ( D ) → A ( C ⊗ ? D )just constructed is inverse to the canonical homomorphism A ( C ⊗ ? D ) → A ( C ) ⊗ ? A ( D )one observes that this is the case by construction after restriction to the algebraic tensorproducts.In order to show Proposition 7.9 in general we must extend Lemma 7.11 from C ∗ -categorieswith finitely may objects to arbitrary C ∗ -categories. Our argument for this will depend onthe following lemma. Lemma 7.12.
1. The functor ⊗ max on C ∗ Alg nu preserves filtered colimits in each argument.2. The functor ⊗ min on C ∗ Alg nu preserves filtered colimits in each argument whosestructure maps are isometric inclusions. C ∗ -categories below. Proof of Prop. 7.9 assuming Lem. 7.12. If C is in C ∗ Cat nu , then we have a canonicalisomorphism colim C (cid:48) C (cid:48) ∼ = −→ C , where the colimit runs over the full subcategories of C with finitely many objects. Thestructure maps of this system are fully faithful functors which are injective on objects. ByLemma 6.3 we therefore get an isomorphism colim C (cid:48) A ( C (cid:48) ) ∼ = → A ( C ) , (7.6)where the structure map of the system of A ( C (cid:48) ) are isometric inclusions. We now considerthe diagram colim C (cid:48) , D (cid:48) A ( C (cid:48) ) ⊗ ? A ( D (cid:48) ) ∼ =Lem. 7 . (cid:47) (cid:47) ∼ = (cid:15) (cid:15) colim C (cid:48) , D (cid:48) A ( C (cid:48) ⊗ ? D (cid:48) ) (cid:15) (cid:15) A ( C ) ⊗ ? A ( D ) (cid:47) (cid:47) A ( C ⊗ ? D )where the left vertical isomorphism uses the assumed Lemma 7.12 and (7.6) (also for D ),and the right vertical arrow exists by the universal property of the colimit. One checksthat the dotted arrow defined by this square is inverse to the canonical homomorphismsince this is true after restricting to the algebraic tensor products for the uncompletedversion A alg of the functor A .In the following discussion we show a couple of results which prepare the actual proof ofLemma 7.12. At the end we use the material in order to derive some additional resultswhich will be used, e.g. in [BEL]. Our presentation will be selfcontained with one exception:the exactness of the maximal tensor product for C ∗ -algebras [BO08, Prop. 3.7.1], but thisdoes not go into the proof of Lemma 7.12.We start with an explicit model for filtered colimits in C ∗ Cat nu . We consider a smallfiltered category I and a functor C : I → C ∗ Cat nu . In the following construction togetherwith Proposition 7.14 we provide an explicit model for the C ∗ -category colim I C . Construction 7.13.
The colimit colim I ∗ Cat nu C C of the image of the diagram in ∗ Cat nu C has the following explicit description. For every i in I we let ι i : C i → colim I ∗ Cat nu C C denote the canonical map.1. objects: The set objects of colim I ∗ Cat nu C C is given by colim I Ob( C ).91. morphisms: For every two objects ¯ C and ¯ C (cid:48) in colim I ∗ Cat nu C C we can (since 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 ∗ Cat nu C 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) )) , (7.7)where I i/ denotes the slice category of objects under i in I , and the colimit is takenin Vect C .3. composition and involution: These structures are defined in the canonical manner.We now define a norm on colim I ∗ Cat nu C C as follows. If ¯ f : ¯ C → ¯ C (cid:48) is any morphism in colim I ∗ Cat nu C C , then there exists i and C, C (cid:48) in C i as in Point 2 above and ( i → i (cid:48) ) in I i/ and a morphism f in C i (cid:48) such that ι i (cid:48) ( f ) = ¯ f . We then define (cid:107) ¯ f (cid:107) := lim ( φ : i (cid:48) → i (cid:48)(cid:48) ) ∈ I i (cid:48) / (cid:107) φ ( f ) (cid:107) C i (cid:48)(cid:48) . Since the map ( φ : i (cid:48) → i (cid:48)(cid:48) ) (cid:55)→ (cid:107) φ ( f ) (cid:107) C i (cid:48)(cid:48) is decreasing and bounded below by 0 the limitexists. Since I is filtered the right-hand side does not depend on the choices of i (cid:48) and f .We form the completion D := colim ∗ Cat nu C I C with respect to the norm defined above. This amounts to forming the completion of themorphism spaces and extending the composition and the involution by continuity. Notethat this process also involves forming the quotient by the subcategory of morphismswith zero norm, and hence the map from the original category to its completion is notnecessarily injective. Since the norm in C i satisfies the C ∗ -equality and inequality for every i in I we conclude from the construction that also the norm on D has these properties.Consequently, D is an object in C ∗ Cat nu .The family of structure maps ( ι i ) i ∈ I provides the first map of the composition C → colim ∗ Cat nu C I C → D (7.8)in Fun ( I , ∗ Cat nu C ), where − stands for forming the constant I -diagram on − . The secondmorphism is induced by the inclusion of the colimit into its completion. Since the inclusionfunctor C ∗ Cat nu → ∗ Cat nu C is fully faithful, the inclusion (7.8) is a morphism C → D in Fun ( I , C ∗ Cat nu ), and hence by adjunction corresponds a functor σ : colim I C → D . (7.9)The following proposition shows that D is an explicit model for the C ∗ -category colim I C . Proposition 7.14.
The functor σ from (7.9) is an isomorphism. roof. We construct an inverse. We have a canonical functor κ : colim ∗ Cat nu C I C → colim I C which in view of the formula (6.6) is just the completion map. Using the notation fromPoint 7.13.2 we have (cid:107) κ ( ¯ f ) (cid:107) colim I C = (cid:107) κι i (cid:48)(cid:48) φ ( f ) (cid:107) colim I C ≤ (cid:107) φ ( f ) (cid:107) C i (cid:48)(cid:48) for all ( φ : i (cid:48) → i (cid:48)(cid:48) ) ∈ I i (cid:48) / . By considering the limit over I i (cid:48) / we conclude that (cid:107) κ ( ¯ f ) (cid:107) colim I C ≤ (cid:107) ¯ f (cid:107) D . This shows that κ extends by continuity to a functor κ : D → colim I C which is necessarilyinverse to σ .As a first application of Proposition 7.14 we show Lemma 7.17 below. Its specialization to C ∗ -algebras has been used in the proof of Proposition 3.8.Let I be a small filtered category. Definition 7.15.
We say that I is countably filtered if for every functor J → I from acountable category the inclusion J → I extends to the cone over J . Example 7.16. If A is a C ∗ -algebra, then the poset of all separable subalgebras of A iscountably filtered. If J is a subset of this poset, then we can extend the inclusion to thecone over J by sending the cone tip to the separable subalgebra (cid:83) j ∈ J A j of A , where theclosure is taken in A .Let C : I → C ∗ Cat nu be a diagram indexed by a small filtered category I , i be in I , and D be a subcategory of C i . Lemma 7.17.
Assume:1. D has countably many objects and separable morphism spaces.2. The composition D → C i ι i −→ colim I C is zero.3. I is countably filtered.Then there exists a morphism φ : i → i (cid:48) in I such that the composition D → C i φ ( i → i (cid:48) ) −−−−→ C i (cid:48) is zero. roof. Using the Assumption 1 on D we can choose a countable set of morphisms M in D such that M ∩ Hom D ( D, D (cid:48) ) is dense for every pair of objects
D, D (cid:48) in D . By thedescription of the norm in colim I C given by Proposition 7.14 and Assumption 2, forevery m in M and n in N we can find φ m,n : i → i m,n in I i/ such that (cid:107) φ m,n ( m ) (cid:107) C i (cid:48)(cid:48) ≤ n .We let J be the subcategory of I with the set of objects { i m,n | m ∈ M, n ∈ N } and thenon-identity morphisms φ m,n : i → i (cid:48) . Using that I is countably -filtered by Assumption3 we can now extend the inclusion of J into I to the cone over J such that the cone tipis sent to an object i (cid:48) of I . We let φ : i → i (cid:48) be the unique morphism which factorizes as i φ m,n −−−→ i m,n → i (cid:48) for every m in M and n in N such that the second morphism belongs tothis extension.For every m in M and n in N we have by construction C ( φ )( m ) = 0. By the densityassumption on M this implies that C ( φ ) annihilates all morphisms of D .As a second application of Proposition 7.14 we prove that filtered colimits preserve isometricinclusions. Let I be a very small filtered category and let ( A → B ) : I → C ∗ Cat nu be anatural transformation of functors. Corollary 7.18. If A i → B i is isometric for every i in I , then the induced morphism colim I A → colim I B is isometric.Proof. This immediately follows from the explicit description of the norm on the colimitsgiven in Proposition 7.14.Next we show that filtered colimits preserve exact sequences of C ∗ -categories. Let I be asmall filtered category and consider a diagram of exact sequences0 → A → B → C → C ∗ Cat nu indexed by I . Lemma 7.19.
The sequence → colim I A → colim I B → colim I C → is exact.Proof. By the definition of an exact sequence in C ∗ Cat nu we have bijections Ob( A ) ∼ =Ob( B ) ∼ = Ob( C ). For a set X we let 0[ X ] denote the C ∗ -category with the set of objects94 and only zero morphisms. We write the diagram of exact sequences as a diagram ofsquares A (cid:47) (cid:47) (cid:15) (cid:15) B (cid:15) (cid:15) B )] (cid:47) (cid:47) C in C ∗ Cat nu which are cartesian and cocartesian. The colimit of cocartesian squares colim I A (cid:47) (cid:47) (cid:15) (cid:15) colim I B (cid:15) (cid:15) colim I B )] (cid:47) (cid:47) colim I C (7.10)is again cocartesian, where we exploit that the two functors 0[ − ] : Set → C ∗ Cat nu andOb( − ) : C ∗ Cat nu → Set are both left adjoints by [Bun, Lem. 3.8.1 & 3.8.2] and thereforecommute with colimits in order to calculate the lower left corner in the colimit. This inparticular implies that the image of the functor colim I A → colim I B is the kernel of thefunctor colim I B → colim I C . In order to show that the square in (7.10) is also cartesianit therefore suffices to show that colim I A → colim I B is isometric. This is exactly theassertion of Corollary 7.18 which is applicable here since the inclusions A i → B i areisometric for all i in I .The following technical lemma is used in the proof of Lemma 7.12. Lemma 7.20.
The minimal tensor product on C ∗ Alg nu preserves isometric inclusions.Proof. If A (cid:48) → A is an isometric inclusion, then we must show that A (cid:48) ⊗ min B → A ⊗ min B is again an isometric inclusion. We choose faithful representations of α and β of A and B as above, respectively. Then we can use α | A (cid:48) as a faithful representation of A (cid:48) . Theassertion is now clear from (7.3).We can now prove Lemma 7.12 and hence complete the proof of Proposition 7.9. Wediscuss the cases ? = min and ? = max separately. Proof of Lem. 7.12 in the case ? = min . Let I be a filtered category, A : I → C ∗ Alg nu bea diagram, and B in C ∗ Alg nu . Then we consider the canonical map colim I ( A ⊗ min B ) → ( colim I A ) ⊗ min B . (7.11)We must show that it is an isomorphism.Since the structure maps of the diagram A are assumed to be isometric inclusions it followsfrom the explicit description of the colimit given by Proposition 7.14 that the canonical95aps A i → colim I A are isometric inclusions. By the Lemma 7.20 the homomorphisms A i ⊗ min B → colim I A ⊗ min B are isometric inclusions, too. Similarly, the structure maps ofthe system A ⊗ min B are isometric inclusions, and hence A i ⊗ min B → colim I ( A ⊗ min B ) isan isometric inclusion. This implies by the Proposition 7.14 that the canonical map (7.11)is an isometry. Since its image clearly contains the dense subset colim I ∗ Alg nu A ⊗ alg B weconclude that it is an isomorphism. Proof of Lem. 7.12 in the case ? = max . In this argument we use that ⊗ max preservesexact sequences of C ∗ -algebras in each argument, see e.g. [BO08, Prop. 3.7.1]. Let I be afiltered category, A : I → C ∗ Alg nu be a diagram, and B in C ∗ Alg nu . Then we considerthe canonical map colim I ( A ⊗ max B ) → ( colim I A ) ⊗ max B . (7.12)We must show that it is an isomorphism.We first consider the special case that B is in C ∗ Alg and A : I → C ∗ Alg . The lattercondition means that A i is unital for every i , and for every morphism i → i (cid:48) in I thestructure map A i → A i (cid:48) preserves units. To prove that (7.12) is an isomorphism in thiscase, it suffices to show that every homomorphism colim I ( A ⊗ max B ) → T for every T in C ∗ Alg factorizes over a homomorphism ( colim I A ) ⊗ max B → T .Let ρ : colim I ( A ⊗ max B ) → T be a homomorphism. It determines a compatible family ofhomomorphisms ( ρ i : A i ⊗ max B → T ) i ∈ I . Using the unit of B we construct a compatiblefamily of homomorphisms ( π i : A i → T ) i ∈ I by π i ( a ) := ρ i ( a ⊗ B ). This family induces ahomomorphism π : colim I A → T . For every i in I we can construct a homomorphism κ i : B → T by κ i ( b ) := ρ i (1 A i ⊗ b ). Because I is connected, these homomorphisms areindependent of i . We will just write κ ( b ) := κ i ( b ) for any choice. Note that ρ i ( a ⊗ b ) = π i ( a ) κ ( b ) for all i in I and a in A i , b in B . Then we get a map ρ (cid:48) : ( colim I A ) ⊗ alg B → T determined by ρ (cid:48) ( a ⊗ b ) := π ( a ) κ ( b ). By the universal property of the maximal tensorproduct ρ (cid:48) extends to ρ (cid:48)(cid:48) : ( colim I A ) ⊗ max B → T . The homomorphism ρ factorizes over ρ (cid:48)(cid:48) as desired.We now consider the case of a diagram A : I → C ∗ Alg nu but still assume that B is unital.We have a split unitalization exact sequence 0 → A → A + → C →
0. Applying to thissequence − ⊗ max B we get again a diagram of exact sequences. We consider the diagram0 (cid:47) (cid:47) colim I ( A ⊗ max B ) (cid:47) (cid:47) ! (cid:15) (cid:15) colim I ( A + ⊗ max B ) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) B (cid:47) (cid:47) (cid:47) (cid:47) ( colim I A ) ⊗ max B (cid:47) (cid:47) ( colim I A + ) ⊗ max B (cid:47) (cid:47) B (cid:47) (cid:47) A + i → C by tensoring with B . The middle verticalmap is an isomorphism by the unital case of this lemma shown above. We now argue thatthe lower horizontal sequence is exact, which will imply that the arrow marked by ! is anisomorphism. We have an equivalence C ∗ Alg nu (cid:39) −→ C ∗ Alg / C given by A (cid:55)→ ( A + → C ),96nd whose inverse is given by ( φ : B → C ) (cid:55)→ ker ( φ ). Furthermore, the canonical functor C ∗ Alg / C → C ∗ Alg preserves colimits in view of the adjunction(( A → C ) (cid:55)→ A ) : C ∗ Alg / C (cid:29) C ∗ Alg : ( B (cid:55)→ ( B ⊕ C → C )) . Hence colim I A ∼ = ker ( colim I ( A + → C )) ∼ = ker ( colim I A + → C )and we have the split exact sequence0 → colim I A → colim I A + → C → . We finally use that − ⊗ max B preserves exact sequences.We finally allow B to be non-unital. Then we get a diagram0 (cid:47) (cid:47) colim I ( A ⊗ max B ) (cid:47) (cid:47) ! (cid:15) (cid:15) colim I ( A ⊗ max B + ) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) colim I A (cid:47) (cid:47) (cid:47) (cid:47) ( colim I A ) ⊗ max B (cid:47) (cid:47) ( colim I A ) ⊗ max B + (cid:47) (cid:47) colim I A (cid:47) (cid:47) colim I A ) ⊗ max − preserves exact sequences.The middle vertical morphism is an isomorphism by the case considered above since B + is unital. The upper horizontal sequence is again exact by Proposition 7.19. We againconclude that the arrow marked by ! is an isomorphism.Next we show that the maximal tensor product on C ∗ Cat nu preserves exact sequences.This generalizes the well-known fact in the case of C ∗ Alg nu , see e.g. [BO08, Prop. 3.7.1].Let 0 → A → B → C → C ∗ Cat nu , and let D be in C ∗ Cat nu . Proposition 7.21.
The sequence → A ⊗ max D → B ⊗ max D → C ⊗ max D → is exact.Proof. On the one hand we will use that the functor A : C ∗ Cat nu i → C ∗ Alg nu preservesexact sequences [Bun, Prop. 8.8]. On the other hand, we will use that A can also be usedto detect exactness. Applying A to the exact sequence (7.13) and using that this functorpreserves exact sequences we first conclude that0 → A ( A ) → A ( B ) → A ( C ) → C ∗ Alg nu . Since the maximal tensor product in C ∗ Alg nu preservesexact sequences we conclude that0 → A ( A ) ⊗ max A ( D ) → A ( B ) ⊗ max A ( D ) → A ( C ) ⊗ max A ( D ) → → A ( A ⊗ max D ) → A ( B ⊗ max D ) → A ( C ⊗ max D ) → B, B (cid:48) of B and D, D (cid:48) of D the sequence0 → Hom A ⊗ max D (( B, D ) , ( B (cid:48) , D (cid:48) )) → Hom B ⊗ max D (( B, D ) , ( B (cid:48) , D (cid:48) )) → Hom C ⊗ max D (( B, D ) , ( B (cid:48) , D (cid:48) )) → C ∗ -categories are bijective on objects so that we can interpret, e.g., B also as an object of A or C .In a similar manner we can deduce that the tensor products in C ∗ Cat nu preserve certainsmall filtered colimits. The following proposition partially generalizes Lemma 7.12 from C ∗ -algebras to C ∗ -categories. Let I be a small filtered category. Let C : I → C ∗ Cat nu bea diagram whose structure maps are injective on the level of sets of objects, and let D bein C ∗ Cat nu . Proposition 7.22.
1. The canonical morphism colim I ( C ⊗ max D ) → ( colim I C ) ⊗ max D (7.14) is an isomorphism.2. If in addition the structure maps of the diagram are isometric inclusions, then colim I ( C ⊗ min D ) → ( colim I C ) ⊗ min D (7.15) is an isomorphism.Proof. By Lemma 6.3 we can commute A with this special kind of colimit. Then we get A ( colim I ( C ⊗ ? D )) Lem. 6 . ∼ = colim I A ( C ⊗ ? D ) Prop. 7 . ∼ = colim I ( A ( C ) ⊗ ? A ( D )) Lem. 7 . ∼ = ( colim I A ( C )) ⊗ ? A ( D ) Lem. 6 . ∼ = A ( colim I C ) ⊗ ? A ( D ) Prop. 7 . ∼ = A ( colim I C ⊗ ? D ) . For the application of Lemma 7.12 in the case ? = min we use Lemma 6.2 to see that thestructure maps of the diagram A ( C ) are isometric inclusions, too. Now arguing as in theproof of Lemma 7.21 we remove A to conclude that (7.14) and (7.15) are isomorphisms.98 emark 7.23. We do not know whether in Proposition 7.22 the assumption that thestructure maps of the diagram are injective on objects is really necessary.
A. Applications to assembly maps
We now introduce a KK-valued version of the Davis–L¨uck assembly map by specializingthe general constructions from [BEa, Sec. 17]. We furthermore explain its relation withthe classical assembly map appearing in the Baum–Connes conjecture, thereby previewingsome of the results from [BEL].We start with C in Fun ( BG, C ∗ Cat ) and let (cid:96) be the localization map from (6.11). Welet j G : BG → G Orb denote the fully faithful inclusion of BG into the orbit category of G which sends the unique object of BG to the orbit G . We then form the left Kan extension j G ! (cid:96) ( C ) as indicated by the dotted arrow in the diagram BG (cid:96) ( C ) (cid:47) (cid:47) j G (cid:36) (cid:36) C ∗ Cat ∞ .G Orb j G ! (cid:96) ( C ) (cid:56) (cid:56) We compose this left Kan extension with the functor kk C ∗ Cat , ∞ from (6.12) (for the trivialgroup G ) and obtain the functor k G C := kk C ∗ Cat , ∞ ( j G ! (cid:96) ( C )) : G Orb → KK (A.1)which is an instance of the functor in [BEa, Def. 17.3]. By Elmendorf’s theorem thisfunctor determines a KK-valued equivariant homology theory H ( − , k G C ) : G Top → KK , X (cid:55)→ H ( X, k G C ) (A.2)on G -topological spaces.We can calculate the values of the functor k G C in (A.1) explicitly. We consider a subgroup H of G . By the point-wise formula for the left Kan extension, the equivalence BH (cid:39) BG /G/H ,by [Bun, Thm. 7.8] (expressing the colimit over BH in terms of the maximal crossedproduct), and using Proposition 6.13 we get the equivalences k G C ( G/H ) (cid:39) kk HC ∗ Cat , ∞ ( j G ! (cid:96) ( C )( G/H )) (A.3) (cid:39) kk C ∗ Cat , ∞ ( colim BH (cid:96) ( C )) (cid:39) kk C ∗ Cat ( C (cid:111) max H ) (cid:39) kk HC ∗ Cat ( C ) (cid:111) max H where we omitted to write Res GH at various places.99et F be a family of subgroups of G and denote by G F Orb the full subcategory of G Orb of G -orbits with stabilizers in F . Then the Davis–L¨uck assembly map for the family F and the functor k G C is defined byAsmbl F ,k G C : colim G F Orb k G C → k G C ( ∗ ) . (A.4)Expressed in terms of the homology theory H ( − , k G C ) in (A.2) this map is equivalent tothe map Asmbl F ,k G C : H ( E F G, k G C ) → H ( ∗ , k G C )induced by the map E F G → ∗ , where E F G is a G - CW -complex representing the homotopytype of the classifying space of G for the family F .In the following we explain the relation of the assembly map (A.4) with the classicalassembly map appearing in the Baum–Connes conjecture [BCH94]. The details of thiscomparison will be developed in [BEL].The constructions in [BEL] depend on the choice of a pair ( C , K ) of a unital C ∗ -category C with G action and an invariant ideal K . We let K u in Fun ( BG, C ∗ Cat ) denote theinvariant full subcategory of K of unital objects. We furthermore consider the category K ( G )std in Fun ( BG, C ∗ Cat nu ) defined in [BEL, Def. 2.8]. Below we follow the conventionsalso used in [BEb], [BEL] to indicate the dependence of certain functors on the pair ( C , K )by just using the symbol C .We use the definition K C ∗ Cat := KK( C , kk C ∗ Cat , ∞ ( − )) : C ∗ Cat ∞ → Sp for the K -theory functor for C ∗ -categories. If we insert k G K u from (A.1) into ΣKK( C , − )we get the functor K G, top C := Σ K C ∗ Cat ( j G ! (cid:96) ( K u )) : G Orb → Sp . We introduce the shift in order to stay compatible with the conventions in [BEL]. On theother hand, we can consider the functor K G, an C := Σ K G, lfkk GC ∗ Cat ( K ( G )std ) : G Fin
Orb → Sp (A.5)obtained by specializing the coefficients of the functor in (1.13) at kk GC ∗ Cat ( K ( G )std ) andapplying the suspension functor Σ.If H is a finite subgroup of G , then we have an equivalence (we again omitted Res GH at100arious places) K G, top C ( G/H ) def (cid:39) ΣKK( C , k G K u ( G/H )) (A.6) (A.3) (cid:39)
ΣKK( C , kk HC ∗ Cat ( K u ) (cid:111) max H ) . . (cid:39) ΣKK H ( C , K u ) (1.21) (cid:39) ΣKK G ( C ( G/H ) , K u ) def (cid:39) Σ K G, lfkk GC ∗ Cat ( K u ) ( G/H ) . The right-hand side of (A.6) looks similar to the evaluation of (A.5) at
G/H , the differencelies in the coefficient categories K u and K ( G )std , respectively. But as a consequence of thegeneral Paschke duality theorem which will be shown in [BEL] one can indeed show thatthese functors are equivalent under some additional mild assumptions on the pair ( K , C ).In the following we explain this in more detail. The comparison involves a third functor K G C : G Orb → Sp defined by specializing [BEa, Def. 17.19] to the case Hg = Σ K C ∗ Cat . We will use the samenotation as in [BEL]. Note that in the present situation K is an ideal in C which impliesby [BEa, Rem. 17.11] that K u is hereditarily additive [BEa, Def. 17.10] and therefore theimplicit assumption going into [BEa, Def. 17.19].The notation for the functors K G, top C and K G C which is used in [BEa] is Σ K C ∗ Cat G K u , max andΣ K C ∗ Cat G K u ,r , respectively. Because K C ∗ Cat is Morita invariant, by [BEa, Prop. 17.21.2]we have an equivalence ( K G, top C ) | G Fin
Orb → ( K G C ) | G Fin
Orb . (A.7)For the following we assume that K is compatible with orthogonal sums [BEL, Def. 17.5].Then the following equivalence is the left vertical equivalence in [BEL, (14.24)]:( K G C ) | G Fin
Orb (cid:39) −→ K G, an C (A.8)(note that the r.h.s. is only define on G Fin
Orb ). Combining (A.8) and (A.7) we obtainthe equivalence ( K G, top C ) | G Fin
Orb (cid:39) −→ K G, an C . (A.9)The relation of the assembly map Asmbl Fin ,k G K u from (A.4) with the classical Baum–Connes101ssembly map Asmbl BC is now best explained by the following diagram: (A.10)ΣKK( C , colim G Fin
Orb k G K u ) ΣKK( C , Asmbl
Fin ,kG K u ) (cid:47) (cid:47) ΣKK( C , k G K u ( ∗ )) colim G Fin
Orb K G, top C Asmbl
Fin ,KG, top C (cid:47) (cid:47) (1) (cid:39) (cid:79) (cid:79) (2) (cid:39) (cid:15) (cid:15) K G, top C ( ∗ ) def’s (cid:39) (cid:79) (cid:79) (3) (cid:15) (cid:15) Σ K C ∗ Cat ( K u (cid:111) max G ) (5) (cid:39) (cid:111) (cid:111) (4) (cid:15) (cid:15) colim G Fin
Orb K G C (cid:39) (cid:15) (cid:15) Asmbl
Fin ,KG C (cid:47) (cid:47) K G C ( ∗ ) (cid:39) (cid:15) (cid:15) Σ K C ∗ Cat ( K u (cid:111) r G ) (6) (cid:39) (cid:111) (cid:111) H ( E Fin
G, K G C ) (8) (cid:39) (cid:15) (cid:15) Asmbl
Fin ,KG C (cid:47) (cid:47) H ( ∗ , K G C ) (7) (cid:39) (cid:15) (cid:15) colim W ⊆ E Fin G KK( C ( W ) , K u std ) Asmbl BC (cid:47) (cid:47) Σ K C ∗ Cat ( K ( G )std (cid:111) r G ) , where the colimit in the lower left corner runs over the G -finite subcomplexes of E Fin G .The arrow marked by (1) is up to inserting definitions the canonical map colim G Fin
Orb
KK( C , k G K u ) → KK( C , colim G Fin
Orb k G K u ) . It is an equivalence since kk( C ) is a compact object of KK. The upper square commutes byconstruction. The arrows marked by (2) and (3) are induced by the natural transformation c from [BEa, Prop. 17.21.1]. The corresponding square commutes by the naturality ofthis transformation. The map (4) is induced by the canonical map from the maximal tothe reduced crossed product. The equivalence marked by (5) is obtained by inserting thecalculation (A.3) for G = H into the definitions. The equivalence marked by (6) is justifiedby [BEa, Cor. 17.20]. The square involving these two maps commutes by an inspection ofthe construction of the natural transformation c in [BEa, (17.21)]. The equivalence (7)will be constructed in [BEL]. Finally, the equivalence (8) involves the Paschke duality andwill be constructed in [BEL]. The map Asmbl BC will also be defined in [BEL]. On thelevel of homotopy groups it is equal to the classical Baum–Connes assembly map [BCH94],slightly extended to G - C ∗ -categories as coefficients.The following is the second main result of [BEL]: Theorem A.1.
The lower square in (A.10) commutes after taking homotopy groups.
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