Paschke Categories, K-homology, and the Riemann-Roch Transformation
aa r X i v : . [ m a t h . K T ] O c t Paschke Categories, K-homology and the Riemann-RochTransformation
Khashayar Sartipi ∗ Abstract
For a separable C ∗ -algebra A , we introduce an exact C ∗ -category called the Paschke Category of A , which is completely functorial in A , and show that its K-theory groups are isomorphic tothe topological K-homology groups of the C ∗ -algebra A . Then we use the Dolbeault complex andideas from the classical methods in Kasparov K-theory to construct an acyclic chain complex inthis category, which in turn, induces a Riemann-Roch transformation in the homotopy categoryof spectra, from the algebraic K-theory spectrum of a complex manifold X , to its topologicalK-homology spectrum. Introduction
The main purpose of this paper is to define a Riemann-Roch transformation from the algebraicK-theory spectrum of a complex manifold to its topological K-homology spectrum. The topologicalK-homology spectrum of a manifold can be defined in various ways, but in this paper, we concernourselves with definitions that use the language of C ∗ -algebras, as they provide a natural frameworkfor the Dolbeault complex. For a separable C ∗ -algebra A , the K-homology spectrum of A can bedefined through the K-theory spectrum of the C ∗ -algebra Q ( A ) called the Paschke dual of A [Pas81].However the definition of the Paschke dual depends on the choice of a representation of A and is onlyfunctorial up to homotopy. Here for any separable C ∗ -algebra A we introduce the Paschke category ( D / C ) A of A whose objects are representations of A and morphisms are the quotient of pseudo-localmodulo locally compact operators. Since we are considering all the representations, this categoryis completely functorial in A . We define structure of an exact C ∗ -category on the Paschke categorywhich in particular, makes it a topological exact category, so that by applying Waldhausen’s S · -construction on the Paschke category and considering the fat geometric realization, we obtain afunctor from C ∗ -algebras to the category of spectra. The K-theory groups of the Paschke categoryare defined to be the (stable) homotopy groups of this spectrum.We observe that the ample representations of A form a strictly cofinal subcategory of thePaschke category and through a standard argument, show that the K-theory spectrum of thePaschke category is homotopy equivalent to the K-theory spectrum of the Paschke dual of the C ∗ -algebra A , which gives the K-homology spectrum of A . We also check that the pull-back maps ofthe Paschke category agree with the classically defined ones up to homotopy. This makes Paschkecategories a convenient place to study K-homology of C ∗ -algebras.By translating the arguments into the language of categories, we can replicate the constructionsin bivariant K-theory [Kas80] to show that the Dolbeault complex of a complex manifold X withcoefficients in a holomophic vector bundle E induces an exact sequence in the Paschke category,obtained by considering the L -completions and applying functional calculus with respect to a ∗ Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago L -completion of sections of a bundledepends on the choice of the metric, then so does the exact sequence we obtain out of the Dolbeaultcomplex, even though there are natural isomorphisms on the relatively compact open subsets.Therefore this process only makes sense on a certain category of vector bundles with a choice ofmetric. We show that this process induces an exact functor from that category, to the category ofacyclic chain complexes in the Paschke category ( D / C ) C ( X ) .To obtain the Riemann-Roch transformation, we need to land in the loop space of the K-theoryspectrum of Paschke category. To achieve this, we first note that there is a natural construction ofGrayson [Gra12] in the homotopy category of spectra, from the K-theory spectrum of the categoryof bounded acyclic double chain complexes to the loop space of the K-theory spectrum of theoriginal category. Then we generalize a construction of Higson given in [Hig95] to obtain a naturalfunctor from the category of acyclic chain complexes in the Paschke category to the category ofacyclic double chain complexes in the Paschke category. The composition of these maps give us aRiemann-Roch transform from the K-theory spectrum of a certain category of vector bundles withmetrics, to the loop space of the K-theory spectrum of the Paschke category. When we restrict thistransformation to a relatively compact open subset, then this map factors through the K-theoryspectrum of the original category of vector bundles. Then we can use the descent properties of thetopological K-homology to glue all the maps on the relatively compact open subsets and obtain theRiemann-Roch transformation.A functorial Riemann-Roch transformation of complex analytic spaces was defined by Levy in[Lev87, Lev08]. We leave to a future paper the functoriality of the Riemann-Roch transformation,with respect to proper maps of complex manifolds.This paper is organized as follows. In section 1, for a C ∗ -algebra A , we define the Paschkecategory (and also a variant called the Calkin-Paschke category) of A , as an exact C ∗ -category, andinvestigate its basic properties, including properties of certain subcategories of the Paschke category.In section 2, we replicate Waldhausen’s arguments to prove cofinality for topological categories, thenrepeat a construction of Grayson to obtain a map (in the stable homotopy category) between certainK-theory spectra, and investigate if the same holds for topological categories. Then we generalizea construction of Higson to obtain an exact functor between certain categories of chain complexesin the Paschke category. In section 3, we use the Dolbeault complex of a complex manifold,together with methods commonly used in the bivariant K-theory, to define an exact sequence inthe corresponding Paschke category. This procedure depends on the choice of metric, and we gothrough a careful argument to show that all the choices induce homotopic maps of spectra. Finallyin section 4, we show that the positive K-theory groups of the Paschke category of A are equalto the shifted K-homology groups of the C ∗ -algebra A . We also show that the natural pull-backmaps agree with the classically defined ones, and use the ingredients from the previous sectionsto define the Riemann-Roch transformation. Finally, for a unital C ∗ -algebra A , we will define anatural pairing of the category of normed right projective A -modules, with the Paschke categoryof A , and show that it has the expected properties. In this paper, we are only considering separable
Hilbert spaces and C ∗ -algebras.We will use theletters A , A ′ , B , C , . . . to refer to categories ”somewhat related” to categories of C ∗ -algebras, andletters A , B , P , . . . to other categories. Also, if A, B are two objects in the category A , then we willuse the notation A ( A, B ) to denote the set (or space) of morphisms from A to B in the category A . Also, if no confusion arises, we will write A ( A ) instead of A ( A, A ). We will use O , A , C , . . . torefer to certain sheaves. 2 .2 Acknowledgment First and foremost, I want to thank my advisor, Henri Gillet, for sharing his ideas, and manyvaluable conversations and thoughtful answers to my questions. I would also want to thank UlrichBunke, who pointed out a gap in one of the arguments, and Nigel Higson, for suggesting a shortcutthat simplified some of the arguments. Also I want to thank Ben Antieau, and Pete Bousfield forhelpful comments. C ∗ -Categories and the Paschke Category Let us start with giving a brief history and basic definitions of C ∗ categories . Karoubi first defined Banach Categories in [Kar68]. A good source for this material is [Kar08]. Later C ∗ -categories weredefined in [GLR85]. Another good source for C ∗ -categories is [Mit02]. Definition 1.1.
The category A is called a complex ∗ -category if:A1 For each two objects A, B of A , A ( A, B ) is a complex vector space and composition of arrowsis bilinear.A2 There is an involution antilinear contravariant endofunctor ∗ of A which preserves objects.The image of x under ∗ will be denoted by x ∗ . It follows that each A ( A, A ) is a ∗ -algebrawith identity.A3 For each x ∈ A ( A, B ), x ∗ x is a positive element of the ∗ -algebra A ( A, A ), i.e. x ∗ x = y ∗ y forsome y ∈ A ( A, A ). Furthermore, x ∗ x = 0 implies x = 0.It follows that the mapping A ( A, B ) × A ( A, B ) → A ( A, A ) defined by ( x, y ) x ∗ y is a A ( A, A )-valued inner product on the right A ( A, A )-module A ( A, B ), where A ( A, A ) acts on A ( A, B ) by composition of arrows.A ∗ -category A is called a normed ∗ -category if:A4 Each A ( A, B ) is a normed space and k xy k ≤ k x kk y k .A normed ∗ -category is called a Banach ∗ -category if:A5 Each A ( A, B ) is a Banach space.A Banach ∗ -category is called a C ∗ -category if:A6 For each arrow x of A , k x k = k x ∗ x k .It follows that each A ( A, A ) is a C ∗ -algebra with identity. A6 shows that the norm on a C ∗ -category is uniquely determined by the norms on the C ∗ -algebra A ( A, A ). In fact, we can saymore: Let A be a ∗ -category where each A ( A, A ) is a C ∗ -algebra, then A can be made into a normed ∗ -category satisfying A6 (but not A5 in general) in a unique way by setting k x k = k x ∗ x k / .Of course any C ∗ -algebra with identity can be considered as a C ∗ -category with a single object. Definition 1.2.
Let A , A ′ be C ∗ -categories. Then a functor F : A → A ′ is called a ∗ -functor if itis a linear functor (i.e. F : A ( A, B ) → A ′ ( F ( A ) , F ( B )) is linear for all objects A, B of A .) and also F ( x ) ∗ = F ( x ∗ ) for all morphisms x in A . Definition 1.3. [Mit02, 3.1.] A non-unital category , is a category of objects and morphismssimilar to a category, except that there need not exist an identity morphism 1 ∈ Hom ( A, A ) foreach object A . A non-unital functor F : A → B between (possibly non-unital) categories A , B is a3ransformation similar to a functor, except that there is no condition on the identity morphisms ofthe category A . Similarly, we can define non-unital C ∗ -categories, and ∗ -functors between them. Definition 1.4. [Mit02, 4.2.] Let A be a C ∗ -category, then a C ∗ -ideal I in the category A is (aprobably non-unital) subcategory of A so that: • The subcategory I has the same objects as the category A . • Each morphism set I ( A, B ) is a norm closed subspace of the space A ( A, B ). • The composition of an arrow in the category A with an arrow in the subcategory I is anarrow in the subcategory I .As a result of the definition above we have: Proposition 1.5. [Mit02, 4.7.] Let j ∈ I ( A, B ) be a morphism in the C ∗ -ideal I of the C ∗ -category A . Then the adjoint morphism j ∗ is also a morphism in the ideal I .Also, we can define the quotient A / I to be the category with the same objects as A and withmorphism sets the quotient Banach space ( A / I )( A, B ) = A ( A, B ) I ( A, B ) . This is also a C ∗ -category. Example 1.6.
We will use B to denote the category of Hilbert spaces with bounded operatorsbetween them, which is an additive C ∗ -category as products and coproducts of a finite number ofHilbert spaces is just their direct summand, and also the set of bounded operators between twoHilbert spaces forms an abelian group, as we can add the operators with each other and compositionon both sides is linear.We will denote the C ∗ -ideal of compact operators by K . Example 1.7.
Let A be a C ∗ -algebra, and let A denote a C ∗ -category. Let R ep A ( A ) denote thecategory of representations of A , i.e. a category whose objects are representations ρ : A → A ( H ),where H is an object in A , and whose morphisms between two representations ρ : A → A ( H )and ρ : A → A ( H ) is the Banach space A ( H , H ). Notice that we are not restricting ourattention to unital representations, i.e. we also consider the zero representation, and other non-unital representations.If A is additive, then it is easy to check that R ep A ( A ) is an additive C ∗ -category as well. Definition 1.8.
Let A be a C ∗ -algebra, and let ρ i : A → B ( H i ), be representations of A for i = 1 , T : H → H is called pseudo-local , if ρ ( a ) T − T ρ ( a ) ∈ K ( H , H ) , ∀ a ∈ A ,and T is locally compact , if both ρ ( a ) T, T ρ ( a ) are in K ( H , H ) for all a ∈ A . Definition 1.9.
Let A be a C ∗ -algebra. Then we define the Paschke category of A to be thequotient category ( D / C ) A := D A / C A , where D A is the category of representations ρ : A → B ( H )of A , where the morphisms between two representations are the pseduo-local operators betweenthem, and the C ∗ -ideal C A has the same objects, but the morphisms are locally compact operators.We define the Calkin-Paschke category of A to be the category where the objects are repre-sentations ρ ′ : A → ( B / K )( H ), and morphisms are again the quotient of pseudo-local operatorsmodulo locally compact operators. We denote the Calkin-Paschke category by ( D / C ) ′ A .Note that there is a natural functor ( D / C ) A → ( D / C ) ′ A , which sends a representation ρ : A → B ( H )to ρ ′ : A → B ( H ) → ( B / K )( H ). 4 otation 1.10. From now on, we will use letters such as ρ, ν to refer to objects of the Paschkecategory, and use similar letters with ”primes”, i.e. ρ ′ , ν ′ to refer to objects of the Calkin-Paschkecategories.In case no confusion should arise, instead of writing T ρ ( a ) is compact for all a ∈ A , we willsimply say T ρ is compact, and similarly for ρ ′ . Example 1.11 (See [HR00, 5.3.2.]) . One can generalize the definition above, by introducing a”relative” version. Let A be a C ∗ -algebra and I ⊂ A a C ∗ -ideal. Then for representations ρ i : A → B ( H i ) for i = 1 ,
2, define D A ( ρ , ρ ) to be the same as the above example, and let C I,A ( ρ , ρ ) = { T ∈ D A ( ρ , ρ ) | T ρ ( a ) , ρ ( a ) T ∈ K ( H , H ) , ∀ a ∈ I } . Note that when I ⊂ J then C A ⊂ C J,A ⊂ C I,A , and if I = A , then we recover the definition above.All of the results on the Paschke category also holds for this relative version, however, by theorem9 and excision for K-homology (cf. [HR00, 5.4.5.]), this does not provide any new information. Definition 1.12. [Kar08, 1.1.6.7.] [K +
00, Def 8.] Let A be an additive category. Then A is called pseudo-abelian if for each object H of A and every morphism p : H → H so that p = p , the kernelof p exists.In the case when A is an additive C ∗ -category, and each self-adjoint projection has a kernel,then we say A is weakly pseudo-abelian . Proposition 1.13. [Kar08, 1.1.6.9.] Let A be a (weakly) pseudo-abelian category, let H be anobject of A and let p : H → H be such that p = p (and also p = p ∗ ). Then the object H splits intothe direct sum H = ker( p ) ⊕ ker(1 − p ) . Proposition 1.14. [Kar08, 1.1.6.10.] [K +
00, Thm 9.] Let A be an additive category. Then thereexists a pseudo-abelian category ˜ A , and an additive functor φ : A → ˜ A which is fully faithful and isuniversal among additive functors from A to a pseudo-abelian category. The pair ( φ, ˜ A ) is uniqueup to equivalence of categories. ˜ A is equivalent to the category where objects are pairs ( H, p ) where H is an object in A and p : H → H is a projector (i.e. p = p ), and morphisms between ( H , p ) and ( H , p ) are morphisms f : H → H in A such that f p = p f = f in A . This category is called the pseudo-abelianization of A .The same statement is true for a C ∗ -category and its weakly pseudo-abelian counterpart. Proposition 1.15.
The weak pseudo-abelianization of the C ∗ -category ( B / K ) is naturally isomor-phic to the Calkin-Paschke category ( D / C ) ′ C .Proof. The objects in the Calkin-Paschke category ( D / C ) ′ C can be considered as pairs ( H, ρ ′ (1)) ofa Hilbert space H and a self-adjoint projection p = ρ ′ (1) ∈ ( B / K )( H ), and morphisms ρ ′ → ρ ′ in ( D / C ) ′ C are the pseudo-local operators modulo locally compact ones, i.e. the operators F ∈ ( B / K )( H , H ) so that F p = p F , modulo the ones that F p = 0 = p F . In other words since F (1 − p ) is locally compact, hence F = F p = p F in the Calkin-Paschke category ( D / C ) ′ C .Therefore we have a natural functor ( D / C ) ′ C → ^ ( B / K ).This functor is faithful, because F = F p = p F are all zero in the category ( B / K ) iff F is locallycompact in the Calkin-Paschke category. The functor is also full, because any F ∈ ( B / K )( H , H )that satisfies F p = p F is pseudo-local. Proposition 1.16.
The Calkin-Paschke category ( D / C ) ′ A is a weakly pseudo-abelian category.
5e need the following two lemmas to prove the proposition above.
Lemma 1.17.
Each self-adjoint projection in ( D / C ) ′ A has a representative in D A which is a self-adjoint projection.Proof. Let P ∈ D A ( ρ ′ ) be a representative for a self-adjoint projection in ( D / C ) ′ A . Hence ρ ′ ( P − P ∗ )and ( P − P ∗ ) ρ ′ are compact operators. Set P ′ = ( P + P ∗ ) /
2. Then P ′ is a self-adjoint operator,hence by Weyl-Von Neumann Theorem [HR00, 2.2.5.] there is a diagonal compact perturbationof P ′ , i.e. there exists an operator P with an orthonormal basis { e i } ∞ i =1 of eigenvectors of P for H with complex numbers λ i as eigenvalues so that and P − P ′ is a compact operator. Therefore P − P is in C A ( ρ ′ ).Let I ⊂ N be the set of indices i such that | λ i | < /
2. Now define the bounded operator Q by Q ( e i ) = e i if i / ∈ I , and set Q ( e i ) = 0 otherwise. Evidently, Q is a self-adjoint projection in thecategory B . We want to show that P − Q is in C A ( ρ ′ ).Define D ( e i ) = − λ i e i if i ∈ I , and D ( e i ) = λ i e i otherwise. Notice that D is a bounded diagonaloperator (of norm at most 2). Also ( P − Q ) e i = λ i e i when i ∈ I , and ( P − Q ) e i = ( λ i − e i otherwise. Furthermore ( P − P ) e i = ( λ i − λ i ) e i for all i ∈ N . Therefore ( P − Q )( e i ) = D ( P − P )( e i ) = ( P − P ) D ( e i ) for all i , hence P − Q = D ( P − P ) = ( P − P ) D . But since P − P ∈ C A ( ρ ′ ), then ( P − P ) ρ ′ , ρ ′ ( P − P ) are compact operators. Therefore ( P − Q ) ρ ′ , ρ ′ ( P − Q ) arealso compact, which proves that P − Q ∈ C A and hence Q ∈ D A ( ρ ′ ). Lemma 1.18.
Let T ∈ D A ( ρ ′ , ρ ′ ) be a pseudo-local operator with closed image V ⊂ H . Let V ⊂ H be the orthogonal complement of ker( T ) . Then for i = 1 , , the projections π i : H → V i and the inclusions ι i : V i → H i are pseudo-local operators.Proof. Since T has a closed image, then it induces an isomorphism of Hilbert spaces from V to V .To simplify the notation, denote T ′ = π T ι : V → V . Let S ′ ∈ B ( V , V ) be the inverse to T ′ and set S = S ′ ⊕ V ⊕ V ⊥ = H → V ⊕ V ⊥ = H . We have ST = ι π and T S = ι π . Firstwe show that S is also pseudo-local. This would show that ι i , π i are pseudo-local.Let ρ ′ i = (cid:16) ρ ′ i ρ ′ i ρ ′ i ρ ′ i (cid:17) ∈ ( B / K )( V i ⊕ V ⊥ i ) for i = 1 ,
2. Since T is pseudo-local, we have T ′ ρ ′ − ρ ′ T ′ and T ′ ρ ′ and ρ ′ T ′ are compact . Therefore ρ ′ = S ′ T ′ ρ ′ = 0 and ρ ′ = ρ ′ T ′ S ′ = 0. Also,since ρ ′ i ( a ) ∗ = ρ ′ i ( a ∗ ), then we can say that ρ ′ i , ρ ′ i are zero, for i = 1 ,
2. Therefore ρ ′ S − Sρ ′ =( ρ ′ S ′ − S ′ ρ ′ ) ⊕
0. But ( ρ ′ S ′ − S ′ ρ ′ ) = S ′ T ′ ( ρ ′ S ′ − S ′ ρ ′ ) = S ′ ρ ′ T ′ S ′ − S ′ ρ ′ = 0. Hence S is pseudo-local.We have ι i ρ ′ i − ρ ′ i ι i = ( ρ ′ i , − ( ρ ′ i , ρ ′ i ) = 0. This proves that ι i is pseudo-local, for i = 1 , ρ ′ i = π i ρ ′ i ι i : A → ( B / K )( V i ) is an object of the Calkin-Paschkecategory ( D / C ) ′ A . This follows from adjointness of ι i and π i and ρ ′ i ( ab ∗ ) = π i ρ ′ i ( a ) ρ ′ i ( b ∗ ) ι i = π i ι i π i ρ ′ i ( a ) ρ ′ i ( b ∗ ) ι i = π i ρ ′ i ( a ) ι i ( π i ρ ′ i ( b ) ι i ) ∗ . Proof of Proposition.
Let P ∈ D A ( ρ ′ ) be a representative for a self-adjoint projection in ( D / C ) ′ A ,and let Q be the projection as in proof of lemma 1.17 (we will only use the fact that Q has a closedimage). Let ι be the inclusion of ker( Q ) in H , and let π be the projection onto the kernel. Then wewant to show that ι is a kernel for Q . We clearly have Qι = 0. Also since Q is bounded from below Where in here, we are denoting the operator induced by T ′ in ( B / K )( V , V ) again by T ′ . We abuse notation ina similar way for S ′ , S, T, π i , ι i . Note that this part of the argument only works in the Calkin-Paschke category. In fact, this is the only part ofthe proof of pseudo-abelianness of the Calkin-Paschke category that does not apply to the Paschke category ( D / C ) A .
6n the orthogonal complement of its kernel, then it has closed image. It follows from lemma 1.18that ι is pseudo-local. Let Q ′ denote ”inverse” of Q restricted to orthogonal complement of ker( Q ),i.e. Q ′ sends image of the projection Q isometrically to the orthogonal complement of ker( Q ).Now let F ∈ D A ( ρ ′ , ρ ′ ) be an operator so that QF ∈ C A ( ρ ′ , ρ ′ ). Then we want to show that F factors through ι up to locally compact operators. We have ιπF = F modulo compact operatorsbecause ( Id H − ιπ ) F = Q ′ QF = 0. Also, if we have ρ ′ ( ιG − F ) = 0, then ρ ′ ι ( G − πF ) = 0.Therefore ρ ′ ( G − πF ) = 0. This completes the proof. Definition 1.19.
Let A be an additive category. Then we say that a chain complex . . . T i − −−−→ ρ i T i −→ ρ i +1 T i +1 −−−→ ρ i +2 T i +2 −−−→ . . . is exact if there is a contracting homotopy , i.e. if there are morphisms S i in A from ρ i +1 to ρ i sothat T i − S i − + S i T i = Id ρ i in A .As a result of this definition, every short exact sequence in A is split, hence A is an exactcategory in the sence of Quillen [Qui73, Sec 2.]. Note that this does not mean all exact sequencesare split.In particular, (using the definition above) the Paschke category ( D / C ) A , the Calkin-Paschkecategory ( D / C ) ′ A , and also D A , B , ( B / K ) are all exact C ∗ -categories.Notice that a map f : A → B of C ∗ -algebras, induces pull-back maps f ∗ : ( D / C ) B → ( D / C ) A and also f ∗ : ( D / C ) ′ B → ( D / C ) ′ A of categories, by precomposing with the representation. This mappreserves exact sequences, hence the pull-back functor is exact, and this process is functorial. We start this subsection by giving a definition similar to [Wal85] . Definition 1.20.
Let B be an additive category. Then a full additive subcategory A is called cofinal if for every object B of the category B , there is an object B ′ in B so that B ⊕ B ′ is isomorphic toan object in A . If we can always take B ′ to be an object in A , then A is called strictly cofinal .In case the category B is exact, we require the subcategory A to be exact as well.Let us recall some definitions and useful properties of representations. Definition 1.21.
A representation ρ : A → B ( H ) of a C ∗ -algebra is called non-degenerate if ρ ( A ) H is a dense subset of H (or equivalently, it is the whole H , cf. [HR00, 1.9.17.].). Anotherequivalent definition is that for each h ∈ H, h = 0, there exists an a ∈ A so that ρ ( a ) h = 0.A representation ρ : A → B ( H ) is called ample if it is non-degenerate, and also for each a ∈ A, a = 0, ρ ( a ) is not a compact operator. Proposition 1.22.
Let Q A denote the full subcategory of ( D / C ) A whose objects are ample repre-sentations, together with the zero representation A → . This is an exact strictly cofinal subcategoryof ( D / C ) A .Let A be a unital C ∗ -algebra and let Q ′ A denote the full subcategory of ( D / C ) ′ A whose objectsare unital injective representations, together with the zero representation A → . This is an exactstrictly cofinal subcategory of ( D / C ) ′ A . This was originally defined for Waldhausen categories by considering coproduct instead of the direct sum. In thispaper, we will only apply the definition to certain Waldhausen categories. roof. Note that direct sum of two non-degenerate representations is non-degenerate, and directsum of a non-degenerate and an ample representation is ample. Given some representation ρ : A → B ( H ), let H = ρ ( A ) H ⊂ H , π : H → H be the orthogonal projection onto the closed subspace,and let ι : H → H be the inclusion. Then we can define ρ : A → B ( H ) by ρ = πρι . Since π, ι are adjoints to each other, and ιπρ = ρ = ριπ then ρ is indeed a representation. Also thesetwo representations are isomorphic as objects of ( D / C ) A , as π, ι are pseudo-local and induce theisomorphism. But ρ is a non-degenerate representation. Hence for any object ρ of ( D / C ) A , andany ample representation ρ of A , ρ ⊕ ρ is isomorphic to an ample representation in ( D / C ) A .If the C ∗ -algebra A is unital and ρ ′ : A → ( B / K )( H ) is an object of ( D / C ) ′ A , then by lemma 1.17 ρ ′ (1) has a representative π ∈ B ( H ) which is a self-adjoint projection. By repeating the argumentabove, ρ ′ is isomorphic to the unital representation ρ ′ : A → ( B / K )( H ), where H ⊂ H is imageof π . Also, direct sum of an injective representation ρ ′ with any representation ρ ′ is injective.Another important property of ample representations, is the following corollary of Voiculescu’stheorem [Voi76], which we mention similar to as stated in [HR00, 3.4.2.]. Theorem 1.23 (Voiculescu) . Let A be a unital C ∗ -algebra, let ρ : A → B ( H ) be a non-degeneraterepresentation, and let ν ′ : A → ( B / K )( H ′ ) be an object in ( D / C ) ′ A . Assume that for each a ∈ A with ρ ( a ) ∈ K ( H ) , we have ν ′ ( a ) = 0 . Then there exists an isometry V : H ′ → H so that V ∗ ρ ( a ) V − ν ′ ( a ) = 0 for all a ∈ A . An important corollary is the following:
Corollary 1.24.
Let ρ , ρ be two ample representations of the C ∗ -algebra A . Then there is aunitary operator U : H → H so that U ρ ( a ) U ∗ − ρ ( a ) is compact for all a . In other words, any two ample representations in the Paschke category are isomorphic. Hence wecan denote the isomorphism class of automorphisms of an ample representation ρ of A by Q ( A ),which is also known as the Paschke dual . Remark . The natural map ( D / C ) A → ( D / C ) ′ A is fully faithful, and by Voiculescu’s theorem,each object ν ′ of ( D / C ) ′ A has an admissible monomorphism to an object which lifts to a non-degenerate representation of A . Since ( D / C ) ′ A is weakly pseudo-abelian, therefore by Voiculescu’stheorem, the full subcategory of the Calkin-Paschke category ( D / C ) ′ A consisting of objects whichlift to the Paschke category ( D / C ) A is cofinal. In this subsection, we recall some standard facts about K-theory and fix our notation for the rest ofthe section. Aside from Waldhausen’s original paper [Wal85], a good source for more informationis [Wei].
Definition 2.1. [Seg74, A.] Let X · be a simplicial space. Then define the topological space k X · k called the fat geometric realization of X · , as the quotient a n X n × ∆ ntop / ∼ + We are abusing the notation for the class of V ∗ ρ ( a ) V in ( B / K )( H ′ ). ∼ + is generated by ( x, f ∗ p ) ∼ + ( f ∗ x, p ) for x ∈ X n , p ∈ ∆ mtop , whenever f :[ m ] → [ n ] in the simplex category ∆ is a face (injective) map.Let X · be a simplicial set (or a discrete simplicial space). Then define the topological space | X · | called the geometric realization of X · , as the quotient a n X n × ∆ ntop / ∼ where the relation ∼ is generated by ( x, f ∗ p ) ∼ + ( f ∗ x, p ) for x ∈ X n , p ∈ ∆ mtop , for any morphism f : [ m ] → [ n ] in the simplex category ∆. Remark . The two definitions of the geometric realization above are equivalent for discretesimplicial spaces.Also, for a simplicial space X · , there is a natural quotient map k X · k → | X · | .The notion of fat geometric realization is better suited for simplicial topological spaces thanthe usual notion, as it takes the topological structure into account. In particular we have theproposition below. Proposition 2.3. :[Seg74, A.1.] Let X · , Y · be simplicial topological spaces.1. If each X n has the homotopy type of a CW-complex, then so does k X · k .2. If X · → Y · is a simplicial map such that X n → Y n is a weak homotopy equivalence for each n , then k X · k → k Y · k is also a weak homotopy equivalence.3. k X · × Y · k is weakly homotopy equivalent to k X · k × k Y · k . Let us recall the general process of defining the algebraic K-theory spectrum of a small
Wald-hausen category ( A , w ) (for more details, see[Wal85]). Definition 2.4.
Let A be a Waldhausen category. Define the simplicial category S · A as follows.First, consider the category of ordered pairs of integers ( j, k ) with 0 ≤ j ≤ k ≤ n that has a uniquemorphism from ( j, k ) to ( j ′ , k ′ ) iff j ≤ j ′ and k ≤ k ′ . Then the objects in S n A are the functors A from this category of pairs to the category A , so that A ( j, j ) = 0 and A ( j, k ) A ( j, l ) ։ A ( k, l )is a cofibration sequence in A whenever 0 ≤ j ≤ k ≤ l ≤ n . The morphisms in S n A are the naturaltransformations A → A ′ , and the weak equivalences are the morphisms that A ( j, k ) → A ′ ( j, k )are all weak equivalences in A . The cofibrations are the morphisms that A ( j, k ) → A ′ ( j, k ) are allcofibrations, and A ( j, l ) ` A ( j,k ) A ′ ( j, k ) → A ′ ( j, l ) are also cofibrations in A whenever 0 ≤ j ≤ k ≤ l ≤ n . Note that a morphism f : [ n ] → [ m ] in the opposite simplex category ∆ op induces a functor S n A → S m A , which sends the object ( j, k ) A ( j, k ) of S n A to the object ( r, s ) A ( f ( r ) , f ( s ))in S m A . This defines a simplicial structure on S · A .Let wS · A be the simplicial category obtained by only considering the weak equivalences in S · A ,and form the nerves in each degree, which yeilds a bisimplicial set N · wS · A . Define the algebraic K-theory spectrum K alg ( A ) of the discrete Waldhausen category A as the spectrum whose n ’th spaceis the goemetric realization | N · w S · S · . . . S · | {z } n times A| . (Or we could have defined the algebraic K-theory space to be the loop space Ω | N · wS · A| . In fact, they have the same (stable) homotopy groups, hencewe may sometimes use the space instead of the spectrum.)By [Mit01], we can define the toplogical K-theory spectrum K top ( A ) of the topological Wald-hausen category A similar as above, i.e. as a spectrum whose n ’th space is the fat geometricrealization k N · w S · S · . . . S · | {z } n times Ak . This description is taken from [TT90, 1.5.1.]. efinition 2.5. Let A be a topological Waldhausen category. Let A δ denote the discrete Wald-hausen category obtained by forgetting the topological structure. There is a natural exact functor A δ → A which induces a natural map K top ( A δ ) → K top ( A ). By remark 2.2, there is a naturalequivalence of K-theory spectra K alg ( A δ ) ∼ = K top ( A δ ).Hence there is a natural comparison map c : K alg ( A δ ) → K top ( A ). Notation 2.6. If A is a topological Waldhausen category, we will simply write K alg ( A ) instead of K alg ( A δ ).Recall from [Wal85, 1.3.]: Definition 2.7.
Let A , B be Waldhausen categories. Then we say that a sequence F → F → F of exact functors from A to B and natural transformations between them is a short exact sequenceof functors , if for each object A of A we have a cofibration sequence F ( A ) F ( A ) ։ F ( A ) in B . Theorem 2.8 (Additivity Theorem) . [Qui73, Sec 3.] [Wal85, 1.3.2.] Let A , B be Waldhausencategories, and let F F ։ F be a short exact sequence of functors from A to B . Then F ∗ , F ∗ + F ∗ : K ( A ) → K ( B ) are homotopic to each other. By [Mit01, 4.2.], the same holds forthe topological Waldhausen categories. Definition 2.9 (Relative K-theory Space) . [Wal85, 1.5.] Let A , B be Waldhausen categories,and let F : A → B be an exact functor. Then define the category [ A F −→ B ] · by [ A F −→ B ] n = S n A × S n B S n +1 B . There is a natural simplicial structure on [ A F −→ B ] · , and the Waldhausencategory structures of S · A , S · B induce one on [ A F −→ B ] in a natural way. Proposition 2.10. [Wal85, 1.5.5.] There are natural functors of Waldhausen categories
B → [ A F −→ B ] · → S · A which in turn induce the homotopy fibration sequence wS · B → wS · [ A F −→ B ] · → wS · S · A . By [Mit01, 4.4.], the same holds for topological Waldhausen categories.
Definition 2.11. [Wal85, After 1.5.3.] Let A , B , C be Waldhausen categories, then a functor F : A × B → C is biexact if for each object A of A and B of B , the functors F ( A, − ) and F ( − , B ) areexact, and also for each cofibration A A ′ in A and B B ′ in B , the map below is a cofibrationin C . F ( A, B ′ ) ∪ F ( A,B ) F ( A ′ , B ) → F ( A ′ , B ′ ) . Proposition 2.12. [Wal85, After 1.5.3.] A biexact functor F : A × B → C of Waldhausen cate-gories, induces a map of bisimplicial categories wS · A ∧ wS · B → wwS · S · C which in turn induces amap of K-theory spectra K ( A ) ∧ K ( B ) → K ( C ) . The same holds for the topological categories [Mit01, 2.8.].
Definition 2.13.
We say that a (topological) Waldhausen subcategory A of B is closed underextensions if for each cofibration sequence in B where the source, and the quotient are in A , thenthe target is isomorphic to an object in A . 10 roposition 2.14. [Wal85, 1.5.9.] If A is a strictly cofinal (topological) Waldhausen subcategoryof B , then the natural map K ( A ) → K ( B ) is a homotopy equivalence. If A is a only a cofinal(topological) Waldhausen subcategory of B which is also closed under extensions, then the naturalmap of K-theory spectra K ( A ) → K ( B ) induces an isomorphism on the i ’th homotopy group when i ≥ . The above statement was originally proved for discrete categories, however, in here we will needto apply it to a certain cofinal subcategory of the Paschke category. The proof goes through fortopological categories with no change, but for the sake of completeness, we repeat the argumenthere.
Notation 2.15.
Let A , B denote topological Waldhausen categories and let F : A → B be an exactfunctor. Then we denote the space of objects in S · A by s · A , and denote the space of objects in[ A F −→ B ] by [ s ( A F −→ B )] · . Beware that the second notation is not standard. Lemma 2.16. [Wal85, 1.4.1.] Let F : A → B be an exact functor of topological Waldhausencategories. Then there is an induced map s · F : s · A → s · B . An isomorphism between two suchfunctors F , F induces a homotopy between s · F , s · F .Proof. The first statement is clear (cf. [Mit01, Page 6.]). To prove the second part, we will explicitlywrite down a simplicial homotopy. Simplicial objects in a category C can be considered as functors X : ∆ op → C , and maps of simplicial objects are natural transformations of such functors. Simplicialhomotopies can be described similarly; namely let ∆ / [1] be denote the category of objects over [1]in the simplex category, i.e. objects are maps [ n ] → [1]. For any X : ∆ op → C , let X ∗ denote thecomposited functor (∆ / [1]) op → ∆ op X −→ C ([ n ] → [1]) [ n ] X [ n ]Then a simplicial homotopy of maps may be identified with a natural transformation X ∗ → Y ∗ .Now, suppose there is a functor isomorphism from F to F given by F : A × [1] → B . Therequired simplicial homotopy then is a map from ([ n ] → [1]) s n A to ([ n ] → [1]) s n B given by( a : [ n ] → [1]) (( A : Ar [ n ] → A ) ( B : Ar [ n ] → B ))where B is defined as the composition Ar [ n ] ( A,a ∗ ) −−−−→ A × Ar [1] Id × p −−−→ A × [1] F −→ B and p : Ar [1] → [1] is given by (0 , , (0 , , (1 , Corollary 2.17. [Wal85] An equivalence of Waldhausen topological categories
A → B inducesa homotopy equivalence s · A → s · B . Therefore if weak equivalences of A are the isomorphisms,denoted by i , then s · A → iS · A is a homotopy equivalence. The first part of this corollary is clear consequence of the lemma. The second part is a resultof considering the simplicial object [ m ] → i m S · A , the nerve of iS · A in the i -direction, and notingthat i S · A = s · A and that face and degenaracy maps are homotopy equivalences by the first partof the corollary. 11 roof of Propositoin 2.14. To prove that a strictly cofinal topological Waldhausen subcategory A of B and B have homotopy equivalent K-theory spaces (and similarly spectra), it suffices to showthat the relative K-theory category wS · [ A ֒ → B ] · is contractible. By property 2 of fat geometricrealization, it suffices to show that each wS n [ A ֒ → B ] · is contractible. Consider the inclusions S n A ֒ → S n B . Then wS n [ A ֒ → B ] · is equivalent to w [ S n A ֒ → S n B ]. But it is easy to show that S n A is a strictly cofinal subcategory of S n B : take an object { B jk } ≤ j Let X · be a spectrum. For n ∈ Z ≥ , let V n X · denote the n ’th stage of thepostnikov filtration of X · , obtained by killing the stable homotopy groups π m ( X · ) for m < n . Inparticular, V X · is the connective part of X · .Following [Wal85, Sec 1.6.], let ( A , w ) be a (topological) Waldhausen category with the subcat-egory w A (sometimes abbreviated to just w ) of weak equivalences. If ( A , v ) is also a (topological)Waldhausen category with weak equivalences so that w is a subcategory of v , then let A v denotethe full (topological) Waldhausen subcategory of ( A , w ) whose objects A are the ones with theproperty that ∗ → A is in v . Recall that for a (topological) category A with cofibrations, if i A denotes the subcategory of isomorphisms, then ( A , i ) is a (topological) Waldhausen category.For a (topological) exact category A , let C A denote the category of chain complexes in A and let Ch A be the category of acyclic chain complexes in A , both of which have chain maps asmorphisms. The categories C A , Ch A have a natural (topological) exact structure; a sequence iscalled exact iff it is exact degreewise. This means the cofibrations are the morphisms which aredegree-wise cofibrations (admissible monomorphisms) and the weak equivalences i are the degree-wise isomorphisms . We introduce a different structure of a (topological) Waldhausen categoryon C A by defining the cofibrations to be the degree-wise cofibrations again, and define the weakequivalences to be the quasi-isomorphisms , which we denote by q . Note that quasi-isomorphismsare considered with respect to embedding the exact category A into an abelian category. Thisdefinition does not depend on the choice of the embedding if A either supports long exact sequences [Gra12, 1.4.], or if A satisfies the condition in [TT90, 1.11.3.]. These conditions are both satisfied if A is a (topological) pseudo-abelian category, cf [TT90, 1.11.5.] and [Gra12, 4.]. Evidently, ( C A , q )is a (topological) Waldhausen category, and Ch ( A ) is equal to ( C A ) q . Furthermore, denote the fullsubcategories of bounded chain complexes and bounded acyclic chain complexes by C b A and Ch b A respectively. Again we have ( C b A ) q = Ch b A . Definition 2.20. [Gra12, 3.1.] Let A be a (topological) exact category. We define a binary chaincomplex in A to be a chain complex in A with two differentials instead of one, i.e. a pair of chaincomplexes with the same objects but possibly different differentials, called the top differential ofthe top chain complex, and the bottom differential of the bottom chain complex. A binary chaincomplex is acyclic if both the top and the bottom chain complexes are acyclic. If we denote a binarychain complex by ( A · , d · , d · ), then A · are the objects of the complex, d · are the top differentials,and d · are the bottom differentials. Let B A and Bi A be the (topological) category of binary chaincomplexes in A and acyclic binary chain complexes in A . Also denote the (topological) categoryof bounded binary chain complexes and the category of bounded acyclic binary chain complexes in A by B b A and Bi b A , respectively. A morphism between two (respectively, acyclic) binary chaincomplexes is a map between the underlying objects which is a chain map with respect to both chain The stable homotopy category (cf. [AA95]) can be considered as the localization of the category of spectra atthe weak homotopy equivalences. In particular, all homotopic maps are equivalent to each other in the homotopycategory, and homotopy equivalences are invertible. We will abuse notation and denote the class of isomorphisms of different categories by i . B A , Bi A , B b A , Bi b A have a natural (topological) exact struc-ture given by exactness at each degree. This means the cofibrations are degree-wise cofibrations,and weak equivalences i are the degree-wise isomorphisms. We can define another structure of a(topological) Waldhausen category on B A , B b A with the cofibrations being the degree-wise cofibra-tions and the weak equivalences being the quasi-isomorphisms which we again denote by q . Thisagain may depend on the choice of embedding A in an abelian category, but does not depend on thatchoice if A is a (topological) pseudo-abelian category. Hence we have a (topological) Waldhausencategory ( B A , q ), and again ( B A ) q , ( B b A ) q are the categories Bi A , Bi b A , respectively.Let us denote the morphism that sends a chain complex ( A · , d · ) to the binary chain complex( A · , d · , d · ) by ∆ : C A → B A , and denote the morphisms that send a binary chain complex torespectively the top and the bottom chain complex by ⊤ , ⊥ : B A → C A . These are exact functors,and we use the same notation for their restriction to C b A → B b A , Ch A → Bi A , B b A → C b A and Bi b A → Ch b A . Let τ, τ b denote the category of maps f in B A and B b A respectively, such that ⊤ f is in qC A , qC b A , respectively, and let β, β b denote the category of maps f in B A and B b A , suchthat ⊥ f is in qC A , qC b A , respectively. Define F : ( C A , q ) → ( B A , τ ) by F ( A · , d · ) = ( A · , d · , ⊤ ◦ F is the identity functor on C A , and F ◦ ⊤ is an exact endofunctor of( B A , τ ).Recall from definition 2.9 that for an exact functor F : A → B between (topological) Waldhausencategories, we have the relative K-theory space denoted by [ A F −→ B ]. We have the followingproposition from Grayson [Gra12, Sec 7.]: Proposition 2.21. Let A be a discrete exact category. Then there is a natural homotopy equiva-lence of spectra K [( Ch b A , i ) ∆ −→ ( Bi b A , i )] ≃ V Ω K ( A ) In particular, there is a natural isomorphism of K-theory groups K n − [( Ch b A , i ) ∆ −→ ( Bi b A , i )] ∼ = K n ( A ) when n ≥ , and there is a natural map in the homotopy category of spectra τ G A : K ( Bi b A , i ) → Ω K ( A ) . (1)The proposition above uses ingredients such as Waldhausen’s fibration and approximation theo-rems [Wal85, 1.6.4, 1.6.7.], the Gillet-Waldhausen theorem [Gil81, 6.2.], and Thomason’s cofinalitytheorem [TT90, 1.10.1.], which we will check for topological categories in a future paper.Let A be (a topological) an exact category. Recall that by definition of the relative K-theoryspace, there is an exact functor Bi b ( A ) → [ Ch b A ∆ −→ Bi b A ] for a (topological) Waldhausen category A . Now, assuming that the (topological) category A ”supports long exact sequences”, we give aseries of maps in the homotopy category of spectra as follows. Following the proof of [Gra12, 4.3.],we first give a map in the homotopy category of spectra G : K [( Ch b A , i ) ∆ −→ ( Bi b A , i )] → Ω K [( C A , i ) ∆ −→ ( B A , i )] . We have the following commutative diagrams: K ( Ch b A , i ) K ( C b A , i ) K ( Bi b A , i ) K ( B b A , i ) K ( Ch b A , q ) K ( C b A , q ) K ( Bi b A , q ) K ( B b A , q )14y Waldhausen’s fibration theorem [Wal85, 1.6.4.], the squares above are cartesian when the cate-gory A is discrete. Therefore we get the cartesian square below. K [( Ch b A , i ) ∆ −→ ( Bi b A , i )] K [( C b A , i ) ∆ −→ ( B b A , i )] K [( Ch b A , q ) ∆ −→ ( Bi b A , q )] K [( C b A , q ) ∆ −→ ( B b A , q )]In the case of topological exact categories, the square above is still commutative (but not necessarilycartesian). Also the argument below works for topological exact categories as well.The lower left hand corner of the diagram is contractible as each of the two categories in therelative K-theory space are contractible. The map K ( Ch b A , i ) K ∆ −−→ K ( Bi b A , i ) is a homotopyequivalence, because the functor P j : ( C b A , i ) → A which sends a chain complex to the term indegree j is exact, and by induction and the additivity theorem, induces an isomorphism K ( C b A , i ) ∼ = K ( ` Z A ) (cf. [Gil81, 6.2.]). Similarly we can say the same for K ( B b A , i ), and note that ∆commutes with these isomorphisms and the identity map on K ( ` Z A ). Thus the upper right handcorner of the diagram above is also contractible. Therefore we have the following sequence ofnatural maps: K [( Ch b A , i ) ∆ −→ ( Bi b A , i )] ∼ = K h [( Ch b A , i ) ∆ −→ ( Bi b A , i )] → i ∼ ←− K h [( Ch b A , i ) ∆ −→ ( Bi b A , i )] → [( Ch b A , q ) ∆ −→ ( Bi b A , q )] i ∗ −→ Ω K h [( C b A , i ) ∆ −→ ( B b A , i )] → [( C b A , q ) ∆ −→ ( B b A , q )] i ∼ ←− Ω K [( C b A , q ) ∆ −→ ( B b A , q )] . When A is a discrete category, all of the maps above are homotopy equivalences, hence G is ahomotopy equivalence. Note that if the fibration theorem holds for topological exact categories,then ∗ (and therefore G ) is a homotopy equivalence for topological categories as well.The next step is to define the homotopy equivalence G : Ω K [( B b A , q ) ⊤ −→ ( C b A , q )] → K [( C b A , q ) ∆ −→ ( B b A , q )] . This map is induced by the commutative diagram of [Gra12, 4.5.]. To be more precise, G is thecomposition of the following sequence of maps:Ω K h ( B b A , q ) ⊤ −→ ( C b A , q ) i ∼ = Ω K B b A , q ) ( C b A , q ) ⊤ ∼ −→ Ω K ( C b A , q ) ( C b A , q )( B b A , q ) ( C b A , q ) 1∆ 1 ⊤ ∼ −→ Ω K ( C b A , q ) 0( B b A , q ) 0 ∆ ∼ = K h ( C b A , q ) ∆ −→ ( B b A , q ) i . Where we used the fact that ⊤ ◦ ∆ = 1 and K-theory of squares is a generalization of relativeK-theory which was defined in [Gra92, Sec 4.] . The proofs only rely on the additivity theorem, which holds for topological categories. K (cid:0) ( B b A ) τ , q (cid:1) → Ω K [( B b A , q ) ⊤ −→ ( C b A , q )], which is a homotopy equivalence in the case of discrete categories. Instead we define thehomotopy equivalence below for the (topological) exact category A . G : K [( B b A , q ) ⊤ −→ ( C b A , q )] → K [( B b A , q ) → ( B b A , τ )] . Notice that we have the following commutative diagram, where each row is a cofiber sequence ,and F : ( C b A , q ) → ( B b A , τ ) is defined by F ( A · , d · ) = ( A · , d · , K ( B b A , q ) K ( C b A , q ) K [( B b A , q ) ⊤ −→ ( C b A , q )] K ( B b A , q ) K ( B b A , τ ) K (cid:2) ( B b A , q ) → ( B b A , τ ) (cid:3) K ⊤ KF ∃ G This induces the desired map G . Similar to [Gra12, 4.8.], we can argue that for the (topological)exact category A , the maps KF, K ⊤ are inverses to each other up to homotopy. (The argumentrelies on the fact that weak equivalence between two functors induces a homotopy between thecorresponding maps of K-theory [Wal85, 1.3.1.], which also holds for topological categories; see[Seg68, 2.1.].) Hence KF is a homotopy equivalence for the (topological) exact category A , whichmeans that G is also a homotopy equivalence.Let ( C b A ) x be the subcategory of chain complexes ( A · , d · ) in C b A , whose euler characteristic χ ( A · ) = P n ( − n A n is equal to zero. When A is a discrete category, according to [Gra12, 5.8.],as a corollary of Thomason’s cofinality theorem [TT90, 1.10.1.], we have a homotopy equivalence K (cid:0) ( C b A ) x , q (cid:1) ∼ −→ V K ( C b A , q ). Then for the discrete exact category A , one has the followingsequence of homotopy equivalences:Ω K (cid:0) ( B b A ) τ , q (cid:1) ∼ = Ω K (cid:0) ( B b A ) β , q (cid:1) = Ω K (cid:0) ( B b A ) β , τ (cid:1) ∼ −→ Ω K (cid:0) ( C b A ) x , q (cid:1) ∼ −→ Ω V K ( C b A , q ) ∼ ←− Ω V K ( A ) ∼ = V Ω K ( A ) (2)where the first homotopy equivalence is given by interchanging the top and the bottom differentials;the second is done by observing that (( B b A ) β , q ) = (( B b A ) β , τ ); the third map is induced by thefunctor ⊤ : (( B b A ) β , τ ) → (( C b A ) x , q ) (note that this is well-defined since ( A · , d · , d · ) in (( B b A ) β , τ )is sent to ( A · , d · ), but acyclicity of ( A · , d · ) shows that the euler characteristic is zero.), which bytheorem [Gra12, 5.9.] is a homotopy equivalence for discrete categories ; the fourth one by [Gra12,5.8.], is a corollary of Thomason’s cofinality theorem [TT90, 1.10.1.]; the last one is true for anyspectrum; and finally, the fifth map is induced by the inclusion A → C b A as the chain complexconcentrated in degree zero, which is a homotopy equivalence is by [Gil81, 6.2.] (Also see [TT90,1.11.7.]) . However, since the map is going in the opposite direction, the homotopy equivalencedoes not necessarily induce a map for topological exact categories.The proof of [Gil81, 6.2.] relies on Waldhausen’s fibration theorem as well. The author willcheck whether this holds for topological exact categories in a future work. Lemma 2.22. Let A be a topological exact category, that ”supports long exact sequences”, andassume that K ( A ) → K ( C b A , q ) is a homotopy equivalence, where the map is induced by inclusionas the chain complex concentrated in degree zero. Then the sequence of maps in 2, composed withthe natural map V Ω K ( A ) → Ω K ( A ) (given by definition of Postnikov tower) factors through K [( B b A , q ) → ( B b A , τ )] . Recall that fibration and cofiber sequences are the same in the category of spectra. The proof relies on Waldhausen’s approximation theorem [Wal85, 1.6.7.] whose proof is quite long! We need the extra assumption [TT90, 1.11.3.] for this theorem to hold, however we are assuming that A ”supportslong exact sequences”, which ensures that there is no problem. A does not ”support long exact sequences”.The pseudo-abelianization (cf. proposition 1.14) ˜ A of the (topological) exact category A inherits(both the topological and) the exact structure of A . Also A embeds in ˜ A as a cofinal subcategory.Hence K n ( A ) → K n ( ˜ A ) is an isomorphism when n > n = 0. Similar to[Gra12, 6.3.], The induced inclusions Ch b ( A ) ֒ → Ch b ( ˜ A ) and Bi b ( A ) ֒ → Bi b ( ˜ A ) are also cofinal,and by repeating the argument in [Gra12, 6.2.], the natural map from the cofiber of K ( Ch b ( A )) → K ( Bi b ( A )) to the cofiber of K ( Ch b ( ˜ A )) → K ( Bi b ( ˜ A )) is a homotopy equivalence . Again bycofinality, V Ω K ( A ) → V Ω K ( ˜ A ) is a homotopy equivalence.Since ˜ A is pseudo-abelian hence as explained before, ˜ A ”supports exact sequences”, and when A is a discrete category, then there is an induced natural map in the homotopy category of spectra τ G A : K ( Bi b ( A )) → K ( Bi b ( ˜ A )) τ G ˜ A −−→ V Ω K ( ˜ A ) ∼ ←− V Ω K ( A ) . Proof of Lemma. Let ( A · , d · , d · ) be an object of ( B b A , q ) τ . By definition, the top chain complex( A · , d · ) is acyclic. This goes to ( A · , d · , d · ) through the first map in the sequence 2, and the secondmap is the identity. The third map sends it to the top chain complex ( A · , d · ) in (( C b A ) x , q ). Thecomposition Ω K (cid:16) ( C b A ) x , q (cid:17) ∼ Ω V K ( C b A , q ) ∼ Ω V K ( A ) ∼ V Ω K ( A ) → Ω K ( A )is equal to the composition G : Ω K (cid:16) ( C b A ) x , q (cid:17) → Ω K ( C b A , q ) ∼ Ω K ( A ) , where the first map is induced by inclusion of categories, and the second is given by the hypothesisof the lemma.The natural map Ω K (( B b A ) τ , q ) → K [( B b A , q ) → ( B b A , τ )] is induced by inclusion. This sendsthe object ( A · jk , d · ,jk , d · ,jk ) ≤ j ≤ k ≤ n of S n ( B b A ) τ to the pair (cid:16) ( A · jk , d · ,jk , d · ,jk ) ≤ j ≤ k ≤ n , (0) ≤ j ≤ k ≤ n +1 (cid:17) .Now, define the map G : K [( B b A , q ) → ( B b A , τ )] → Ω K ( C b , q ) by G (cid:0) ( A · jk , d · ,jk , d · ,jk ) ≤ j ≤ k ≤ n , ( A ′ , d ′ , d ′ ) n +1 (cid:1) = ( A · jk , d · ,jk ) ≤ j ≤ k ≤ n , where ( A ′ , d ′ , d ′ ) n +1 is an object of S n +1 ( B b A , τ ), the first term is an object of (cid:2) ( B b A , q ) → ( B b A , τ ) (cid:3) and the second term is an object of S n ( C b A , q ). Then use the natural homotopy equivalence k wS · S · Ek ∼ = Ω k wS · Ek for the topological Waldhausen category E = ( C b A , q ). Generalizing a construction given by Higson in [Hig95, Page 6.], for a C ∗ -algebra A we define thefunctor τ HA : C ( D / C ) A → B ( D / C ) A below. Definition 2.23. Let ( ρ · , T · ) be a chain complex in C ( D / C ) A . Define τ HA ( ρ · , T · ) to be the binarychain complex whose n ’th term is the graded object ν n = (cid:0) ⊕ n − i = −∞ ( ρ n − ⊕ ρ n ) (cid:1) ⊕ ρ n in ( D / C ) A ,where the last piece is of degree n . The top differential (temporarily denoted by) ⊤ n from ν n to The reason why the n ’th homotopy groups are isomorphic follows from cofinality when n > 0. But for n = 0 anextra argument is needed. n +1 is a degree 1 map, where its i ’th degree piece from ρ n − ⊕ ρ n (of degree i ) to ρ n ⊕ ρ n +1 (ofdegree i + 1) is the trivial one (i.e. is identity on ρ n and zero on ρ n − .) for i ≤ n − 1, and its n ’thdegree piece is equal to ρ n T n −−→ ρ n +1 . The bottom differential (temporarily denoted by) ⊥ n from ν n to ν n +1 is a degree 0 map, where its i ’th degree piece from ρ n − ⊕ ρ n to ρ n ⊕ ρ n +1 is again thetrivial one (i.e. is identity on ρ n and zero on ρ n − .) for i ≤ n − 1, and its n ’th degree piece is thetrivial inclusion ρ n ( Id, −−−→ ρ n ⊕ ρ n +1 . ... . .. . . . ⊕ ( ρ n − ⊕ ρ n ) ⊕ ( ρ n − ⊕ ρ n ) ⊕ ρ n . . . ⊕ ( ρ n ⊕ ρ n +1 ) ⊕ ( ρ n ⊕ ρ n +1 ) ⊕ ( ρ n ⊕ ρ n +1 ) ⊕ ρ n +1 . . . ⊕ ( ρ n +1 ⊕ ρ n +2 ) ⊕ ( ρ n +1 ⊕ ρ n +2 ) ⊕ ( ρ n +1 ⊕ ρ n +2 ) ⊕ ( ρ n +1 ⊕ ρ n +2 ) ⊕ ρ n +2 ... ... ... ... ρ n +1 ... T n − T n T n +1 It is easy to see that the bottom chain complex is split exact, and the top chain complex isexact iff the original chain complex ( ρ · , T · ) is exact. This process is functorial with respect tochain maps in a trivial way. Finally, note that if we start with a chain complex of length n ,then we will get a binary chain complex of length n + 1. Hence we also have the natural functor τ HA : Ch b ( D / C ) A → Bi b ( D / C ) A . This functor is not exact; however we can tweak the structures ofthe categories to obtain an exact functor. Definition 2.24. Let A be a C ∗ -algebra and let Ch ′ ( D / C ) A , Bi ′ ( D / C ) A denote the categories withthe same objects as Ch b ( D / C ) A , Bi b ( D / C ) A respectively, but with morphisms and exact structurecoming from the category D A . To be precise, a morphism in Ch ′ ( D / C ) A from the chain complex( ρ · , T · ) to ( ν · , S · ) is given by a chain map f n : ρ n → ν n in the category D A , and morphisms in Bi ′ ( D / C ) A are defined similarly as a chain map in D A with respect to both the top and the bottomchain complex. We say a sequence of chain complexes in Ch ′ ( D / C ) A is exact, iff the sequence isexact at each degree in D A , and similarly define the exact structure on Bi ′ ( D / C ) A .There are natural functors Ch ′ ( D / C ) A → Ch b ( D / C ) A and Bi ′ ( D / C ) A → Bi b ( D / C ) A . Thesefunctors are exact, since exactness in D A guarantees exactness in ( D / C ) A . Lemma 2.25. The functor τ HA defined in 2.23 induces an exact functor τ HA : Ch ′ ( D / C ) A → Bi ′ ( D / C ) A . (3) Therefore, we have a natural map of K-theory spectra τ HA : K ( Ch ′ ( D / C ) A ) → K ( Bi ′ ( D / C ) A ) . This is proved by observing that infinite direct sum of identities is equal to identity in D A . Notethat this is not true in the paschke category ( D / C ) A . Proof. Let ( ρ · i , T · i ) denote objects in Ch ′ ( D / C ) A for i ∈ Z , and let f · i : ( ρ · i , T · i ) → ( ρ · i +1 , T · i +1 ) bemorphisms that give an exact sequence in Ch ′ ( D / C ) A , with the degree-wise contracting homotopygiven by g · i : ( ρ · i +1 , T · i +1 ) → ( ρ · i , T · i ). Then we need to show that τ HA ( g · i ) τ HA ( f · i ) + τ HA ( f · i − ) τ HA ( g · i − )is equal to identity at each degree of τ HA ( ρ · i , T · i ). This is true since at degree n this is given byinfinite direct sum g ni f ni + f ni − g ni − and g n − i f n − i + f n − i − g n − i − . But by assumption, each term isequal to identity in D A , hence their infinite direct sum is also equal to identity.18one of this process works in the Calkin-Paschke category ( D / C ) ′ A , as infinite direct sums of ρ ′ : A → ( B / K )( H ) is not necessarily defined, since infinite direct sum of compact operators doesnot have to be compact. In fact if the infinite direct sum of ρ ′ is well-defined, then by an Eilenbergswindle argument we can show that the class corresponding to ρ ′ in K top ( A ) = Ext ( A ) defined in[BDF77] is zero. This section will contain a great deal of computations, and to ease the readability, we will fix someof our notations. Notation 3.1. Fix χ ( t ) = t √ t . The functions χ, φ will be used for functional calculus. Theletter X denotes manifolds, U, V are used for open subsets of the manifold, λ for a partition ofunity, and γ for cutoff functions. The letters D, d will be used for differential operators, and ¯ ∂ willdenote the Dolbeault operator.We will use E for vector bundles, g, h will be reserved for a metric on the manifold, and on thebundle respectively. The letters α, β will be isomorphisms of vector bundles, ϕ, σ, ψ will be mapsof vector bundles.The letter I will be used as a map of Hilbert spaces induced by identity map on a bundle (withdifferent choices of metrics), π will refer to projection onto the L -integrable functions on an opensubset, and ι will denote extension by zero of L sections on an open subset to the whole space. For the definition and basic properties of functional calculus, see the appendix B. One could applyfunctional calculus to an essentially self-adjoint operator, and in certain cases we get interestingproperties. Lemma 3.2. Let X be a differentiable manifold, E a differentiable vector bundle over X , and let D ∈ Diff ( E, E ) be a differential operator of order . Consider the representation ρ : C ( X ) → B ( L ( X, E )) .1. Let φ be a bounded Borel function on R , whose Fourier transform is compactly supported.Then φ ( D ) is a well-defined bounded operator acting on L ( X, E ) , which is in fact, pseudo-local. [HR00, 10.3.5, 10.6.3.]2. Assume in addition that D is an elliptic operator. Let φ ∈ C ( R ) , then φ ( D ) : L ( X, E ) → L ( X, E ) is a locally compact operator. [HR00, 10.5.2.] Now we are ready to define the functor ˆ τ DX . Definition 3.3. Let X be a complex manifold of dimension n , and let E be a holomorphic vectorbundle on X . We will use the Dolbeault complex 16 to define an exact sequence in the Paschkecategory of C ( X ).Fix some hermitian metric g on X and a Hermitian metric h on E and let H i be the spaceof L -integrable sections of the bundle ∧ ,i T ∗ X ⊗ E over X . There are natural representations ρ i : C ( X ) → B ( H i ) given by point-wise multiplication of a function on X with the L -section.Let ¯ ∂ ∗ E be the formal adjoint of the Dolbeault operator ¯ ∂ E (with respect to the metrics g, h .), andconsider the essentially self-adjoint differential operator D E = ¯ ∂ E + ¯ ∂ ∗ E of order 1 [HR00, 11.8.1.].Therefore we can apply functional calculus to D E with respect to the function χ ( t ) = t √ t , to19btain a bounded operator D E √ D E = χ ( D E ) ∈ B ( ⊕ i H i ). By lemma 3.2, this is a pseudo-localoperator with respect to the ρ i ’s, so if χ i ( D E ) = ¯ ∂ i √ D E ) denotes the restriction of χ ( D E ) to B ( H i , H i +1 ), then we have the following chain complex in the Paschke category ( D / C ) C ( X ) .ˆ τ DX,g ( E, h ) : 0 → ρ χ ( D E ) −−−−−→ ρ χ ( D E ) −−−−−→ . . . χ n − ( D E ) −−−−−−→ ρ n → . (4)To show that this is in fact an exact sequence in the Paschke category, we need to find pseudo-localoperators P i : H i +1 → H i which give a contracting homotopy , i.e. P i χ i ( D E ) + χ i − ( D E ) P i − − Id H i is a locally compact operator. It is easy to see if P i = ¯ ∂ ∗ i √ D E : H i +1 → H i , then P i χ i ( D E ) + χ i − ( D E ) P i − − Id H i = D E , which is locally compact by lemma 3.2. This shows that 4 is anexact sequence. Proposition 3.4. Let X be a compact complex manifold and E a holomorphic vector bundle.Then the chain complex 4 considered as a complex of Hilbert spaces and bounded operators, isquasi-isomorphic to the Dolbeault complex 16 with coefficients in E .Proof. It is easy to see that the diagram below commutes:0 A , X ( E ) A , X ( E ) . . . A ,nX ( E ) 00 H H . . . H n ¯ ∂ ¯ ∂ ( D E ) − / ¯ ∂ n − ( D E ) − n/ χ ( D E ) χ ( D E ) χ n − ( D E ) Since the image of the vertical maps are dense, then by the Hodge-decomposition A.5, we can seethis sends Harmonic forms isomorphically to the cohomology of the complex below.The definition 3.3, is not very easy to work with when we restrict to open subsets, becauserestriction of an essentially self-adjoint operator to an open subset is not necessarily essentiallyself-adjoint. We will give an equivalent definition in 3.7 for any symmetric elliptic operator. Lemma 3.5. [HR00, 10.8.4.] Let X be a differentiable manifold, E a differentiable vector bundleon it, U ⊂ X an open subset, and D , D ∈ Diff ( E, E ) are order one essentially self-adjointdifferential operators, so that D | U = D | U . Then if f ∈ C ( U ) , we have ρ ( f ) χ ( D ) − ρ ( f ) χ ( D ) is a compact operator, where ρ : C ( U ) → B ( L ( X, E )) is given by pointwise multiplication, and χ ( t ) = t √ t . Definition 3.6. Let X be a locally compact, and Hausdorf topological space. We say that theopen cover { U j } j is a good cover , if it is countable, locally finite, and each open set U j is relativelycompact. Definition 3.7. [HR00, 10.8.] Let X be a (non-compact) differentiable manifold, E a differentiablevector bundle on X , and let D ∈ Diff ( E, E ) be a symmetric elliptic differential operator of order1. Let { U j } j be a good cover (definition 3.6), and let { λ j } j be a partition of unity subbordinate tothe cover, and let { γ j } j be compactly supported non-negative continuous functions, so that γ j | U j is equal to (the constant function) one. Then the symmetric differential operator D j = γ j Dγ j is supported on a compact set, hence by lemma B.5, is essentially self-adjoint. Therefore if the The operators P i are also called the parametrices ρ : C ( X ) → B ( L ( X, E )) (where the L -completion is of course defined withrespect to a choice of a metric on X and one on E .) is given by pointwise multiplication, then wecan define χ D := X j ρ ( λ / j ) χ ( D j ) ρ ( λ / j ) , (5)as the partial sums are bounded in norm and the series converges in the strong operator topology.One can see that χ D is self-adjoint. χ D depends on the choice of the open cover, the partitionof unity, and the cut-off functions γ j ’s, but if f ∈ C ( X ) is compactly supported, then ρ ( f ) χ D hasonly finitely many terms, hence by lemma 3.5, if D ∈ Diff ( E, E ) is any essentially self-adjointdifferential operator which agrees with D on support of f , then ρ ( f ) χ D − ρ ( f ) χ ( D ) is compact,and in particular if D is itself essentially self-adjoint, then χ D − χ ( D ) is locally compact. Hence thechoices do not matter up to locally compact operators. Therefore, we have a well-defined operator χ D in the Paschke category ( D / C ) C ( X ) . Definition 3.8. Let X be a complex manifold, then denote the category of holomorphic vectorbundles on X by P ( X ). This is an exact category.It is straightforward to show that P ( X ) has a small skeletal subcategory. For each vector bundleon X , there is a set of metrics, hence if we denote the category of holomorphic vector bundles witha choice of metric by P m ( X ), (i.e. objects are pairs ( E, h ) of a holomorphic vector bundle with ahermitian metric, and morphisms are bundle maps) then without loss of generality, we can assumethat this is a small category. This inherits an exact structure from the category P ( X ).Let g be a hermitian metric on X , then denote the bounded category of holomorphic vectorbundles with a choice of metric by P m, b ( X, g ), where objects again are pairs ( E, h ) of a holomorphicvector bundle with a choice of a hermitian metric, and a morphism from ( E , h ) to ( E , h ) is a mapof bundles E → E , so that the induced map L ( X, ∧ , ∗ T ∗ X ⊗ E ) → L ( X, ∧ , ∗ T ∗ X ⊗ E ) is abounded map of Hilbert spaces. We say a sequence . . . → ( E i − , h i − ) → ( E i , h i ) → ( E i +1 , h i +1 ) → . . . is exact in P m, b ( X ), if there are smooth maps of bundles σ i : E i +1 → E i which are a contractinghomotopy, and also the induced maps L ( X, ∧ , ∗ T ∗ X ⊗ E i +1 ) → L ( X, ∧ , ∗ T ∗ X ⊗ E i ) are bounded. This is again a small category.Note that there P m, b ( X ) is a subcategory of P m ( X ), and there is a forgetful map P m ( X ) →P ( X ). Both of these functors are exact. Proposition 3.9. Let X be a complex manifold, and let g be a hermitian metric of boundedgeometry on X . Recall from definition 2.24 that Ch ′ ( D / C ) C ( X ) denotes the category of boundedacyclic chain complexes in ( D / C ) C ( X ) where the exact structure is induced by that of D C ( X ) .The map ˆ τ DX,g defined in 3.3 induces an exact functor from P m,b ( X ) to Ch ′ ( D / C ) C ( X ) . The proof of proposition above, will take the entirety of the next subsection, as we will need toprove a series of technical lemmas (which are probably known to the experts). We include themwith great details for readers who may not have a background in the topic.Throughout this section, we had fixed a single metric on the manifold X and do the rest of ourcomputations. Let us investigate effect of the choice of the metric g on ˆ τ DX,g . Lemma 3.10. Let X be a differentiable manifold, and let E be a differentiable vector bundle on X . Let d ∈ Diff ( E, E ) be a differential operator, so that for each metric g on X and h on E , D = d + d ∗ is an essentially self-adjoint elliptic differential operator. Let g , g be two metrics on X ,and let h , h be metrics on E . Denote d + d ∗ with respect to ( g , h ) , ( g , h ) by D , D respectively. Notice that a smooth contracting homotopy always exists [AA67, 1.4.11.]. The only condition here is boundednessof σ i ’s. hen there is a unitary isomorphism L ( X, E ; g , h ) → L ( X, E ; g , h ) that commutes with χ ( D ) up to locally compact operators.Proof. Let g t = (1 − t ) g + tg , h t = (1 − t ) h + th for 0 ≤ t ≤ 1. Then both g t , h t are metrics (andin case both g , g are hermitian, then so are g t and etc.). Denote the Hilbert space of L -sectionsof E with respect to the metric g t , h t by H t . Let ν t : X → R ≥ be the ”square root of the measure”given by the Radon Nikodym theorem so that dµ t ( Z ) = R Z ν t dµ for each measurable subset Z of X and each t . Let S t : E → E be the square root of the positive definite map E h −→ E ∗ h ∗ t −→ E . ,then T t ( x ) = η ( x ) S t ( x ) acts fiberwise, hence it is pseudo-local. Also for L sections η, ζ in H , h T t η, T t ζ i t = Z X ( h t )( T t η )( T t ζ ) dµ t = Z X ν h t ( S t η )( S t ζ ) dµ t = Z X h t ( S ∗ t S t η )( ζ ) dµ = Z X h t h ∗ t h ( η )( ζ ) dµ = h η, ζ i . Therefore we have unitary maps T t : L ( X, E ) → L ( X, E ) (where the L -completions are withrespect to ( g , h ) , ( g t , h t ), respectively.). Consider the the path t T ∗ t χ ( D t ) T t from χ ( D ) to T ∗ χ ( D ) T . Since χ − ∈ C ( R ), therefore T ∗ t χ ( D t )+12 T t ∈ ( D / C )( ρ ) is a self-adjoint projec-tion up to locally compact operators. Hence by lemma 1.17, without loss of generality we canassume T ∗ t χ ( D t )+12 T t ∈ B ( L ( X, E )) (where the L -completion is with respect to ( g , h ).) is aself-adjoint projection, and by [HR00, 4.1.8.] this path of projections induces a unitary operator W : L ( X, E ; g , h ) → L ( X, E ; g t , h t ) such that W ∗ ( T ∗ χ ( D )+12 T ) W = χ ( D )+12 . Therefore, W T is the unitary isomorphism that commutes with χ ( D ) up to locally compact operators. Notation 3.11. Let X be a differentiable manifold, and let E , E be differentiable vector bun-dles on X . Choose metrics g on X and h , h on E , E . To shorten the notation, we say( X, g ; E , h ; E , h ) is a metric pair .Let X be a complex manifold, and let E , E be holomorphic vector bundles on X . Choose her-mitian metrics g on X and h , h on E , E , and let set D E = ¯ ∂ E + ¯ ∂ ∗ E and D E = ¯ ∂ E + ¯ ∂ ∗ E be thecorresponding Dolbeault operators. To shorten the notation, we say ( X, g ; E , h , D E ; E , h , D E )is a hermitian pair . Definition 3.12. Let ( X, g ; E , h ; E , h ) be a metric pair. We say an operator T (or a familyof operators) is locally bounded with respect to ( X, g ; E , h ; E , h ), if for each relatively compactopen subset U of X , there exists an induced operator T U : L ( U, E | U ) → L ( U, E | U ) so that T U isa bounded linear operator, and if for each pair of relatively compact open subsets U , U , we have π U U ∩ U T U ι U U ∩ U = π U U ∩ U T U ι U U ∩ U where in here π UV : L ( U, E | U ) → L ( V, E | V ) is the projection defined by multiplication by thecharacteristic function of U ∩ V , and ι UV : L ( V, E | V ) → L ( U, E | U ) is extension by zero.Beware that this definition is not exactly the same as more well-known definitions of lo-cal boundedness. Also note that there does not need to be a uniform bound on k T U k . How-ever, in case there is a uniform bound on T U (say M ), then we can ”glue” them to obtain T : L ( X, E ) → L ( X, E ), by simply choosing a relatively compact open neighborhood U of x , and setting T ( ζ )( x ) = T U ( π U ζπ ∗ U )( x ). This is independent of choice of U and k T ( ζ ) k ≤ M k ζ k ,as this holds for almost every point x . Notice that by [Lax07, Page 150.] this exists and varries continuously. Recall that evaluating ζ ∈ L ( X, E ) at a point x ∈ X only makes sense up to subsets of measure zero in X . xample 3.13. Let ( X, g ; E , h ; E , h ) be a metric pair (definition 3.11), and let ϕ : E → E be a continuous bundle map. Then ϕ is locally bounded (definition 3.12). Example 3.14. Let ( X, g ; E, h , E, h ) be a metric pair (definition 3.11), and let L ( X, E ; h i ) = L ( X, E ; g, h i ) denote the space of L -sections of E on X with respect to the metric h i on E (and g on X ). Then the identity map Id : E → E induces a locally bounded map (definition 3.12) from L ( X, E ; h ) to L ( X, E ; h ) which we denote by I ( h , h ) throughout this section. Lemma 3.15. Let ( X, g ; E, h , D E, ; E, h , D E, ) be a hermitian pair (definition 3.11). Then D E, − D E, is locally bounded (definition 3.12).Proof. Recall from definition A.6 that the metric h i can be considered as a linear map of bundlesfrom E to the dual bundle E ∗ , which by abuse of notation we denote with h i again. Let h ∗ i : E ∗ → E denote the dual maps induced by h i , let θ denote the composition E h −→ E ∗ h ∗ −→ E , and let ϑ ∗ denotethe composition E ∗ h ∗ −→ E h −→ E ∗ .Consider f ⊗ e ∈ C ∞ ( X, ∧ , ∗ T ∗ X ⊗ E ), we have: D E, ( f ⊗ e ) − D E, ( f ⊗ e ) = (cid:0) ¯ ∂ E + ( ⋆ ⊗ h ∗ ) ¯ ∂ ( ⋆ ⊗ h ) (cid:1) ( f ⊗ e ) − (cid:0) ¯ ∂ E + ( ⋆ ⊗ h ∗ ) ¯ ∂ ( ⋆ ⊗ h ) (cid:1) ( f ⊗ e )= ( ⋆ ⊗ h ∗ ) ¯ ∂ ( ⋆f ⊗ h ( e )) − ( ⋆ ⊗ h ∗ ) ¯ ∂ ( ⋆f ⊗ h ( e ))= ( ⋆ ⊗ h ∗ ) (cid:0) ¯ ∂ ( ⋆f ) ⊗ h ( e ) + ⋆f ⊗ ¯ ∂h ( e ) (cid:1) − ( ⋆ ⊗ h ∗ ) (cid:0) ¯ ∂ ( ⋆f ) ⊗ h ( e ) + ⋆f ⊗ ¯ ∂h ( e ) (cid:1) = (cid:0) ⋆ ¯ ∂ ⋆ f ⊗ e + ⋆ ⋆ f ⊗ h ∗ ¯ ∂h ( e ) (cid:1) − (cid:0) ⋆ ¯ ∂ ⋆ f ⊗ e + ⋆ ⋆ f ⊗ h ∗ ¯ ∂h ( e ) (cid:1) = ⋆ ⋆ f ⊗ ( h ∗ ¯ ∂ ( e ∗ ) − h ∗ ¯ ∂ ( ϑ ∗ e ∗ ))= ⋆ ⋆ f ⊗ (cid:0) h ∗ ¯ ∂ ( e ∗ ) − h ∗ ( ϑ ∗ ¯ ∂ ( e ∗ ) + e ∗ ¯ ∂ ( ϑ ∗ ) (cid:1) = ⋆ ⋆ f ⊗ θ ( e ) ¯ ∂ ( ϑ ∗ )where in here, e ∗ = h ( e ). The term above does not have any differentials of f ⊗ e ; recall ⋆⋆ is ± k θ k i , k ϑ ∗ k i vary continuously with respect to x ∈ X , and i = 1 , 2, hence the term θ ( e ) ¯ ∂ ( ϑ ∗ ) isbounded with respect to both norms on the relatively compact set U . Lemma 3.16. Let ( X, g ; E, h ; E, h ) be a metric pair (definition 3.11), let and let D i ∈ Diff ( E, E ) be an essentially self-adjoint differential operators with respect to the metric h i for i = 1 , , so that D − D is a locally bounded operator (definition 3.12). Let I ( h , h ) denote the locally boundedmap induced by the identity map of E (example 3.14). Then for each relatively compact open subset U of X , we have π χ ( D ) ι I ( h , h ) U = I ( h , h ) U π χ ( D ) ι in the Paschke category ( D / C ) C ( U ) , where in here, π i : L ( X, E ; h i ) → L ( U, E | U ; h i ) is the pro-jection and ι i is extension by zero, for i = 1 , . This proof closely follows that of [HR00, 10.9.5.]. Proof. Similar to [HR00, 10.3.5.] we argue that if u, v be compactly supported smooth sections of ∧ , ∗ T ∗ X ⊗ E , and φ is a Schwartz function , then since φ ( x ) = π R e √− sx ˆ φ ( s ) ds , then we can pairˆ φ with the smooth function s 7→ h ( I ( h , h ) e √− sD I ( h , h )) u, v i = h e √− sD u, v i to obtain h φ ( D ) u, v i = h I ( h , h ) φ ( D ) I ( h , h ) u, v i = 12 π Z h I ( h , h ) e √− sD I ( h , h ) u, v i ˆ φ ( s ) ds, We can not directly apply this result here, even though they look similar; the problem is that in [HR00, 10.9.5.]it is required for both D , D to be essentially self-adjoint with respect to the same given inner product, which is notthe case here. φ, u, v be as above (with the extra assumption that s ˆ φ ( s ) is a smooth function, also notethat D , D both share the invariant domain of smooth compactly supported functions.), then wehave h ( I ( h , h ) φ ( D ) I ( h , h ) − φ ( D )) u, v i = 12 π Z h ( I ( h , h ) e √− sD I ( h , h ) − e √− sD ) u, v i ˆ φ ( s ) ds. By fundamental theorem of calculus we know that h ( I ( h , h ) e √− sD I ( h , h ) − e √− sD ) u, v i = h ( e √− sD − e √− sD ) u, v i = √− R s h ( I ( h , h ) e √− tD I ( h , h )( D − D ) e √− s − t ) D ) u, v i , and by repeating the argument in [HR00, 10.3.6, 10.3.7.] we obtain that there exists a constant C φ < ∞ (which only depends on φ ) so that k I ( h , h ) φ ( D ) I ( h , h ) − φ ( D ) k ≤ C φ k D − D k .Now, let φ be a normalizing function (i.e. φ − χ ∈ C ( R ).) that satisfies the conditions above,and let φ ǫ ( x ) = φ ( ǫx ). Then φ ǫ is also a normalizing function, and hence φ ǫ ( D i ) − χ ( D i ) is a locallycompact operator for any ǫ > 0. But as ǫ → 0, we get k I ( h , h ) φ ǫ ( D ) I ( h , h ) − φ ǫ ( D ) k = k I ( h , h ) φ ( ǫD ) I ( h , h ) − φ ( ǫD ) k ≤ ǫC φ k D − D k → . In other words, there are elements of equivalency class of locally compact operators equivalent to I ( h , h ) χ ( D ) I ( h , h ) and χ ( D ) respectively, which get arbitrarily close. But these are linear sub-spaces of pseudo-local operators, hence these subspaces have to be the same, i.e. I ( h , h ) χ ( D ) I ( h , h ) − χ ( D ) is locally compact. This finishes the proof. Corollary 3.17. Let ( X, g ; E, h , D E, ; E, h , D E, ) be a hermitian pair (definition 3.11), let I ( h , h ) be the locally bounded map induced by the identity map of E (example 3.14). Let U be a relativelycompact open subset of X , and let π i : L ( X, ∧ , ∗ T ∗ X ⊗ E ; h i ) → L ( U, ∧ , ∗ T ∗ X ⊗ E | U ; h i ) be theprojection and let ι i be its adjoint. Then π χ D E ι I ( h , h ) U = I ( h , h ) U π χ D E ι in the Paschkecategory ( D / C ) C ( U ) . Definition 3.18. Let X be a differentiable manifold, and let E , E be differentiable vector bundleson X . Let α : E → E be a smooth bundle map. Choose metrics g, h , h on X, E , E respectively.We say α preserves the metrics , if the dual map of bundles β : E ∗ → E ∗ on the dual vector bundles(defined by β ( e ∗ )( e ) = e ∗ ( α ( e )).) makes the diagram below commute. E E E ∗ E ∗ . αh h β Lemma 3.19. Let ( X, g ; E , h ; E , h ) be a metric pair (definition 3.11), and let α : E → E be a smooth isomorphism of vector bundles that preserves the metrics (definition 3.18). Let D ∈ Diff ( E , E ) be an essentially self-adjoint differential operator of order one. Then χ ( D ) = α − χ ( αDα − ) α .Proof. Since α preserves the metric on each fiber, then the induced map α : L ( X, E ) → L ( X, E )is a unitary map, i.e. α − = α ∗ . Since D is symmetric, then αDα − = αDα ∗ is also symmetric.24lso if x ∈ Domain( αD ∗ α − ) ⊂ L ( X, E ), then there exists a constant M so that for each y ∈ Domain( αDα − ) ⊂ L ( X, E ), we have |h x, αD ∗ α − y i| ≤ M k y k [HR00, 1.8.2.]. But if x ′ = α − x, y ′ = α − y ∈ L ( X, E ), then this is equivalent to saying that |h x ′ , D ∗ y ′ i| ≤ M k y k , i.e. x ′ ∈ Domain( D ∗ ). Since D is essentially self-adjoint, then x ′ ∈ Domain( D ), hence x ∈ Domain( αDα − ),i.e. αDα − is also essentially self-adjoint. Hence χ ( αDα − ) ∈ B ( L ( X, E )) is well-defined.Assume that the Fourier transform of the bounded Borel function φ is compactly supported,then for small values of s > 0, we have e √− sαDα − = αe √− sD α − . Hence by [HR00, 10.3.5.] itis easy to argue that φ ( αDα − ) = αφ ( D ) α − . Now if φ is a normalizing function, then φ ( D ) − χ ( D ) , φ ( αDα − ) − χ ( αDα − ) are locally compact. Lemma 3.20. Let ( X, g ; E , h , D E ; E , h , D E ) be a hermitian pair (definition 3.11), and let α : E → E be a smooth isomorphism of vector bundles that preserves the metrics. Then αD E α − − D E is locally bounded (definition 3.12).Proof. Recall from definition A.6 that the metrics induce conjugate linear smooth bundle isomor-phisms h i : E i → E ∗ i to the dual bundle, for i = 1 , 2, and h ∗ i : E ∗ i → E i is the inverse. Let β : E ∗ → E ∗ be the map of bundles dual to α . Since α is a smooth isomorphism of vector bundles,then α ¯ ∂ E − ¯ ∂ E α is locally bounded. Therefore by 18, to prove the lemma it suffices to showthat the term below is locally compact, where f ⊗ e is a smooth section of ∧ , ∗ T ∗ X ⊗ E , and e = α − e , e ∗ = h ( e ) ( α ¯ ⋆ E ∗ ¯ ∂ ¯ ⋆ E α − − ¯ ⋆ E ∗ ¯ ∂ ¯ ⋆ E )( f ⊗ e ) = α ¯ ⋆ E ∗ ¯ ∂ (¯ ⋆f ⊗ h ( e )) − ¯ ⋆ E ∗ ¯ ∂ (¯ ⋆f ⊗ h ( e ))= α ¯ ⋆ E ∗ (cid:0) ¯ ∂ (¯ ⋆f ) ⊗ h ( e ) + ¯ ⋆f ⊗ ¯ ∂h ( e ) (cid:1) − ¯ ⋆ E ∗ (cid:0) ¯ ∂ (¯ ⋆f ) ⊗ h ( e ) + ¯ ⋆f ⊗ ¯ ∂h ( e ) (cid:1) = (cid:0) (¯ ⋆ ¯ ∂ ¯ ⋆ ) f ⊗ e + ¯ ⋆ ¯ ⋆f ⊗ αh ∗ ¯ ∂h ( e ) (cid:1) − (cid:0) (¯ ⋆ ¯ ∂ ¯ ⋆ ) f ⊗ e + ¯ ⋆ ¯ ⋆f ⊗ h ∗ ¯ ∂h ( e ) (cid:1) = ¯ ⋆ ¯ ⋆f ⊗ ( αh ∗ ¯ ∂h ( e ) − h ∗ ¯ ∂h ( e ))= ¯ ⋆ ¯ ⋆f ⊗ ( h ∗ β − ¯ ∂β ( e ∗ ) − h ∗ ¯ ∂ ( e ∗ )) But β is also a smooth isomorphism of vector bundles hence ¯ ∂β − β ¯ ∂ is locally bounded. Corollary 3.21. Let → E ϕ −→ E ϕ −→ E → be a short exact sequence of holomorphic vectorbundles on the complex manifold X . Choose a hermitian metric g on X , and h on E . Then weget an induced hermitian metric on the subbundle ϕ ( E ) of E . Extend this metric to hermitianmetric h on all of E . Then there exists a smooth map of bundles σ : E → E which is anisomorphism from E to the orthogonal complement of ϕ ( E ) in E . Let h be the hermitianmetric induced by this isomorhpism.Hence ( X, g ; E ⊕ E , h ⊕ h , D E ⊕ D E ; E , h , D E ) is a hermitian pair (definition 3.11), andwe have a smooth isomorphism ( ϕ , σ ) : E ⊕ E → E . By definition of the metrics, it is easy tocheck that this isomorphism preserves metrics. Therefore as a corollary of 3.20, D E ⊕ D E − ( ϕ , σ ) − D E ( ϕ , σ ) (6) is locally bounded. Corollary 3.22. Let ( X, g ; E , h , D E ; E , h , D E ) be a hermitian pair (definition 3.11), and let α : E → E be a smooth isomorphism of vector bundles on X . Let U be a relatively compact opensubset of X , let π i : L ( X, ∧ , ∗ T ∗ X ⊗ E i ) → L ( U, ∧ , ∗ T ∗ X ⊗ E i | U ) be the projection and let ι i beits adjoint. Then α U π χ ( D E ) ι = π χ ( D E ) ι α U in the Paschke category ( D / C ) C ( U ) , where by abuse of notation, we are denoting the map inducedby α U from L ( U, ∧ , ∗ T ∗ X ⊗ E | U ) → L ( U, ∧ , ∗ T ∗ X ⊗ E | U ) by α U as well. roof. Consider the hermitian metric h ′ on E (defined through the diagram in definition 3.18.)so that the bundle isomorphism α : E → E preserves the metrics. Let D ′ E = ¯ ∂ E + ¯ ∂ ∗ E be theDolbeault operator with respect to h ′ , let π ′ : L ( X, ∧ , ∗ T ∗ X ⊗ E ; h ′ ) → L ( U, ∧ , ∗ T ∗ X ⊗ E | U ; h ′ )be the projection, and let ι ′ be its adjoint. Denote the map induced by α from L ( U, ∧ , ∗ T ∗ X ⊗ E | U ) to L ( U, ∧ , ∗ T ∗ X ⊗ E | U ; h ′ ) by α ′ U , and let I ( h , h ′ ) denote the locally bounded map inducedby the identity of E (example 3.14). Therefore α U = I ( h , h ′ ) U α ′ U and: α U π χ ( D E ) ι = I ( h , h ′ ) U α ′ U π χ ( D E ) ι = I ( h , h ′ ) U π ′ α ′ χ ( D E ) ι = I ( h , h ′ ) U π ′ χ ( αD E α − ) α ′ ι By lemma 3.19= I ( h , h ′ ) U π ′ χ ( D ′ E ) α ′ ι By lemma 3.20 and [HR00, 10.9.5.]= I ( h , h ′ ) U π ′ χ ( D ′ E ) ι ′ α ′ U = π χ ( D E ) ι I ( h , h ′ ) U α ′ U By corollary 3.17= π χ ( D E ) ι α U . Remark . One may wonder if we can change the metric g on X in the corollary above as well.Consider the case where E = E is the trivial bundle of rank one, α is the identity map, and h = h . When g , g are two different hermitian metrics on X , the symbols of ¯ ∂ + ¯ ∂ ∗ g , ¯ ∂ + ¯ ∂ ∗ g are not equal to each other, and there is no indication on why after applying functional calculus,we should get the same operator in the Paschke category. However for a relatively compact opensubset U , the operator induced by identity I ( g , g ) U : L ( U, E | U ; g , h ) → L ( U, E | U ; g , h ) isthe identity on the underlying vector spaces (although these Hilbert spaces are different as they havedifferent inner products.), hence I ( g , g ) should not induce a map between the chain complexesˆ τ DX,g ( E , h ) , ˆ τ DX,g ( E , h ). Proof of proposition 3.9. We have already defined ˆ τ DX,g on the objects of the category, and showedthat ˆ τ DX,g ( E, h ) is an exact sequence in the Paschke category ( D / C ) C ( X ) . We need to show functo-riality and exactness. Before going further, let us fix some notation.Let ϕ : ( E , h ) → ( E , h ) be a morphism in P m, b ( X ). Choose a good cover (definition3.6) { U j } j so that for i = 1 , j , there exists an open subset V j of X that containsclosure of U j , and that E i | V j and ∧ , ∗ T ∗ X | V j is isomorphic to the trivial bundle on V j . In otherwords, there exists holomorphic isomorphisms of bundles α j : ∧ , ∗ T ∗ X ⊗ E | V j → V j × C k and β j : ∧ , ∗ T ∗ X ⊗ E | V j → V j × C m where k, m are ranks of the corresponding bundles. Then ψ j = β j ϕ | V j α − j : V j → M m,k ( C ) is a holomorphic matrix valued function. let D i,j = γ j D E i γ j for i = 1 , 2. Let { λ j } j be a partition of unity subbordinate to the cover { U j } j , and let γ j besmooth cutoff functions which are equal to one on U j . Also, let π i,j : L ( X, ∧ , ∗ T ∗ X ⊗ E i ) → L ( U j , ∧ , ∗ T ∗ X ⊗ E i | U j ) be the projection and let ι i,j be its adjoint. For n ∈ Z > , let D n denotethe Dolbeault operator corresponding to the trivial rank n bundle, let π nj : L ( X, X × C n ) → L ( U j , U j × C n ) be the projection, and let ι nj be its adjoint. Then by corollary 3.22, for relativelycompact subset U j of V j , we get that α j,U j π ,j χ D E ι ,j = π kj χ D kj ι kj α j,U j β j,U j π ,j χ D E ι ,j = π mj χ D mj ι mj β j,U j 26n the Paschke category ( D / C ) C ( U j ) . Now, let f ∈ C ( X ) be compactly supported. Then there areonly finitely many of the V j ’s that intersect support of f , i.e. the sums below are all finite. ( ϕχ D E − χ D E ϕ ) ρ ( f ) = ( X j λ / j ϕχ ( D ,j ) λ / j − X j λ / j χ ( D ,j ) ϕλ / j ) ρ ( f )= ( X j λ / j π ,j ϕι ,j π ,j χ ( D ,j ) ι ,j λ / j − X j λ / j π ,j χ ( D ,j ) ι ,j ϕπ ,j λ / j ) ρ ( f )= ( X j λ / j β − j,U j ψ j α j,U j π ,j χ ( D ,j ) ι ,j λ / j − X j λ / j π ,j χ ( D ,j ) ι ,j β − j,U j ψ j α j,U j π ,j λ / j ) ρ ( f )= ( X j λ / j β − j,U j ψ j π kj χ D kj ι kj α j,U j λ / j − X j λ / j ι ,j β − j,U j π mj χ D mj ι mj ψ j α j,U j π ,j λ / j ) ρ ( f )= ( X j ι ,j β − j,U j π mj χ D mj ψ j ι kj α j,U j λ j − X j ι ,j β − j,U j π mj χ D mj ι mj ψ j α j,U j π ,j λ j ) ρ ( f ) . Where the last equality holds because, in the first sum χ D kj is pseudo-local, and hence up tocompact operators, commutes with multiplication by the matrix valued continuous function λ / j ψ j that vanishes at infinity, therefore ( λ / j ψ j ) χ D kj − χ D mj ( λ / j ψ j ) is compact for each j ; and in thesecond sum λ / j χ D mj − χ D mj λ / j is also compact for each j , and both sums are finite.We conclude the first part of the proof by noting that ψ j ι kj α j,U j λ j = ι mj ψ j α j π ,j λ j , hence each term in the sum above is zero, and therefore ϕ induces a map from ˆ τ DX,g ( E , h ) toˆ τ DX,g ( E , h ) in the category Ch ′ ( D / C ) C ( X ) .It is straightforward to check that ˆ τ DX,g ( ϕ ◦ ϕ ) = ˆ τ DX,g ( ϕ ) ◦ ˆ τ DX,g ( ϕ ). This shows that ˆ τ DX,g isa functor. Remark . Note that the condition on ϕ : L ( X, ∧ , ∗ T ∗ X ⊗ E ) → L ( X, ∧ , ∗ T ∗ X ⊗ E ) beingbounded is not used in the proof of why ˆ τ DX,g is functorial. Also holomorphicity of ϕ was not neededin the argument above, we only needed continuity to show that multiplication by λ j ψ j commuteswith χ ( D ) modulo compact operators.Now, to prove that ˆ τ DX,g is an exact functor, let0 ( E , h ) ( E , h ) ( E , h ) 0 ϕ ϕ σ σ be an exact sequence in P m,b ( X ). Then by definition of exactness in this category, there existssmooth sections σ : E → E , σ : E → E so that σ ϕ = Id E , ϕ σ = Id E , and similar to ϕ i , σ i also induce a bounded map of Hilbert spaces L ( X, ∧ , ∗ T ∗ X ⊗ E i +1 ) → L ( X, ∧ , ∗ T ∗ X ⊗ E i ),for i = 1 , h ′ be the hermitian metric on E induced by h , h , i.e. h ′ = σ ∗ h σ + ϕ ∗ h ϕ : E → E ∗ where in here, σ ∗ : E ∗ → E ∗ and ϕ ∗ : E ∗ → E ∗ are the dual maps to σ , ϕ respectively. Then thesubbundles ϕ ( E ) , σ ( E ) of E are orthogonal with respect to h ′ , and the induced metrics on thesesubbundles match with the metrics h , h respectively, i.e. the isomorphism between E ⊕ E and E preserves the metric (definition 3.18). Hence by corollary 3.21 ( σ , ϕ ) D ′ E ( ϕ , σ ) − D E ⊕ D E 27s locally bounded, where D ′ E is the Dolbeault operator on E with respect to the metric h ′ .Therefore by lemma 3.19, we get that χ D E ⊕ χ D E = χ D E ⊕ D E = ( ϕ , σ ) χ ( σ ,ϕ ) D ′ E ( ϕ ,σ ) ( σ , ϕ ) . By corollary 3.17, for any relatively compact open subset U of X , we have I ( h ′ , h ) U π χ D E ι = π ′ χ D ′ E ι ′ I ( h ′ , h ) U , where π , π ′ are the projections L ( X, ∧ , ∗ T ∗ X ⊗ E ) → L ( U, ∧ , ∗ T ∗ X ⊗ E | U )with respect to the metrics h , h ′ and ι , ι ′ are their adjoints, respectively. Also I ( h ′ , h ) U : L ( U, ∧ , ∗ T ∗ X ⊗ E | U ; h ) → L ( U, ∧ , ∗ T ∗ X ⊗ E | U ; h ′ ) is the map induced by Id E (example3.14). This factors through L ( U, ∧ , ∗ T ∗ X ⊗ E | U ; h ) ( σ ,ϕ ) −−−−−→ L ( U, ∧ , ∗ T ∗ X ⊗ E | U ) ⊕ L ( U, ∧ , ∗ T ∗ X ⊗ E | U ) ( ϕ ,σ ) −−−−−→ L ( U, ∧ , ∗ T ∗ X ⊗ E | U ; h ′ ) Because ( φ , σ ) has norm one, and ( σ , ϕ ) has a bounded norm (independent of U ), then normof I ( h ′ , h ) U is also independent of U , therefore we can glue all the data to obtain I ( h ′ , h ) χ D E = χ D ′ E I ( h ′ , h ). This proves that ˆ τ DX,g is exact. Lemma 3.25. Let X be a complex manifold, and let U be an open subset. Then the diagram belowcommutes up to homotopy. K ( P b,d ( X, g )) K ( Ch ′ ( D / C ) C ( X ) ) K ( P b,d ( U, g )) K ( Ch ′ ( D / C ) C ( U ) ) res ˆ τ DX,g res ˆ τ DU,g Proof. Let ( E, h ) be an object of P b,d ( X ). It suffices to show that in the diagram below (which is not commutative on the nose), res XU ˆ τ DX,g ( E, h ) is naturally isomorphic to ˆ τ DU,g res XU ( E, h ). P b,d ( X, g ) Ch ′ ( D / C ) C ( X ) P b,d ( U, g ) Ch ′ ( D / C ) C ( U ) res XU ˆ τ DX,g res XU ˆ τ DU,g Denote the restriction map L ( X, ∧ , ∗ T ∗ X ⊗ E ) → L ( U, ∧ , ∗ T ∗ X ⊗ E | U ) given by multiplyingwith the characteristic function of U by π .Let u be compactly supported section of L ( U, ∧ , ∗ T ∗ X ⊗ E | U ). Then by [HR00, 10.3.1.] thereexists ǫ > | s | < ǫ , e √− sD UE u = e √− sD E π ∗ u are supported on U . Let φ be a normalizingfunction so that its Fourier transform is supported in the interval [ − ǫ, ǫ ]. Then by [HR00, 10.3.5.] weget that φ ( D UE ) u = πφ ( D E ) π ∗ u . Since φ − χ ∈ C ( R ), then by lemma 3.2 χ ( D UE ) = πχ ( D E ) π ∗ in thePaschke category ( D / C ) C ( U ) . Therefore π is a chain map from res XU ˆ τ DX,g ( E, h ) to ˆ τ DU,g res XU ( E, h ).Since ππ ∗ = Id and π ∗ π − Id is characteristic function of X \ U which is locally compact in( D / C ) C ( U ) , then π induces an isomorphism.Therefore there is a natural transformation from res XU ˆ τ DX,g to ˆ τ DU,g res XU , meaning these twofunctors induce homotopic maps of K-theory spectra.28 roposition 3.26. Let X be a complex manifold. Then for each relatively compact open subset V of X there exists an exact functor τ DV that makes the square below commute up to homotopy.Furthermore, these functors are compatible with further restriction to open subsets, i.e. for an opensubset W of V , the triangle on the bottom of the diagram commutes up to homotopy as well. K ( P b,d ( X, g )) K ( Ch b ( D / C ) C ( X ) ) K ( P ( X )) K ( Ch ′ ( D / C ) C ( V ) ) K ( Ch ′ ( D / C ) C ( W ) ) ˆ τ DX,g res XU ∃ τ DU ∃ τ DW res UV (7) Proof. For each object E of P ( X ), choose a hermitian metric h ( E ) . Then define τ DV,h ( E ) = res XV ˆ τ DX,g ( E, h ( E )). Also, for a morphism of bundles ϕ : E → E , define τ DV,h ( ϕ ) through thecomposition below, where the first map is given by projection, and the last one is given by extensionby zero. L ( X, ∧ , ∗ T ∗ X ⊗ E ) → L ( V, ∧ , ∗ T ∗ X ⊗ E | V ) ˆ τ DV,g ( ϕ | V ) −−−−−−→ L ( V, ∧ , ∗ T ∗ X ⊗ E | V ) → L ( X, ∧ , ∗ T ∗ X ⊗ E )Note that ˆ τ DX,g ( ϕ ) is not necessarily defined, as ϕ could induce an unbounded map of Hilbert spaces,however by restricting to the relatively compact open subset V , the composition above is indeed awell-defined map.Since ˆ τ DV,g is a functor, then τ DV,h is also a functor, i.e. for composable maps of bundles ϕ , ϕ ,we have τ DV,h ( ϕ ◦ ϕ ) = τ DV,h ( ϕ ) ◦ τ DV,h ( ϕ ). Exactness of τ DV,h also follows from that of ˆ τ DV,g . Hencewe have an induced map of spectra τ DV,h : K ( P ( X )) → K ( Ch ′ ( D / C ) C ( V ) ) . (8)The square in the diagram 7 commutes (up to homotopy) because for any object ( E , h ) of P b,d ( X, g ), by corollary 3.17, the identity map of E induces an isomorphism from res XV ˆ τ DX,g ( E , h )to res XV ˆ τ DX,g ( E , h ( E )) = τ DV,h ( E ). Also for a morphism ϕ : ( E , h ) → ( E , h ) in P b,d ( X, g ), thedifference res XV ˆ τ DX,g ( ϕ ) − τ DV,h ( ϕ ) is locally compact, because multiplying by characteristic functionof X \ V is locally compact in ( D / C ) C ( V ) .The functor defined τ D − ,h commutes (up to homotopy) with restriction to further open subset W ⊂ V because multiplying by characteristic function of V \ W is locally compact in ( D / C ) C ( W ) .Therefore the triangle in the diagram 7 commutes as well.Note that the choices of metrics h ( E ) on E do not affect the map 8 up to homotopy, becauseagain by corollary 3.17, for any two choices h , h , the objects τ DV,h ( E ) , τ DV,h ( E ) are naturallyisomorphic, hence all the different functors are homotopic. Corollary 3.27. Let X be a complex manifold. Then the functor τ D defined in proposition 3.26commutes with restriction to open subsets, i.e. for open subset U of X and relatively compact opensubset V of X and open subset W of U ∩ V which is relatively compact as an open subset of U , thediagram below commutes up to homotopy. Note that we are assuming the axiom of choice. Also, we are only working over a small skeletal subcategory of P ( X ). ( P ( X )) K ( Ch ′ ( D / C ) C ( V ) ) K ( P ( U )) K ( Ch ′ ( D / C ) C ( W ) ) τ DV res XU res VW τ DW Proof. Consider the diagram below, where all the arrows with no labels are the natural ones. K ( P m,b ( X, g )) K ( Ch ′ ( D / C ) C ( X ) ) K ( P m,b ( U, g )) K ( Ch ′ ( D / C ) C ( U ) ) K ( P ( X )) K ( Ch ′ ( D / C ) C ( V ) ) K ( P ( U )) K ( Ch ′ ( D / C ) C ( W ) ) ˆ τ DX,g ˆ τ DU,g τ DV τ DW The squares on the left and the one on the right commute because restriction maps (and theforgetful functors P m,b → P ) are natural. By lemma 3.25 the square on the top commutes up tohomotopy. By proposition 3.26 the squares in the back and on the front commute up to homotopyas well. This proves that the square on the bottom commutes up to homotopy. Remark . All the results in this subsection hold whether we use K alg or K top . In this subsection, we will compute the K-theory groups of the Paschke category and the Calkin-Paschke category.Let A be a C ∗ -algebra. Let K top · ( A ) denote the topological K-homology groups of A , which arecontravariant functors of the C ∗ -algebra, and let K · top ( A ) denote the topological K-theory groupsof A , which are covariant functors. The reason for the unusual naming is that we are primarilyinterested in the case when A is the C ∗ -algebra C ( X ) of continuous complex valued functions onthe (locally compact and Hausdorf) topological space X which vanish at infinity, and in this casefunctoriality matches the expectations.Let A be a topological exact category, and recall that K top ( A ) denotes the K-theory spectrumof A with respect to the fat geometric realization. Since the additivity theorem holds for K-theoryof topological categories, then this is a connective spectrum , i.e. there are no negative K-theorygroups.Here is the main result of this subsection. Theorem 4.1. The (1 − i ) ’th topological K-homology group K − itop ( A ) of a C ∗ -algebra A , is isomor-phic to the i ’th topological K-theory groups of the exact C ∗ -categories ( D / C ) A and ( D / C ) ′ A for i ≥ .If A is unital and nuclear, then K top ( A ) = K top (( D / C ) ′ A ) as well. For a ∗ -morphism f : A → B ,this isomorphisms commutes with respect to the pull-back maps f ∗ . n particular if A = C ( X ) for a locally compact Hausdorf topological space X , then the topolog-ical K-homology groups K topi − ( X ) are isomorphic to K i (( D / C ) ′ C ( X ) ) for i ≥ . This isomorphismalso commutes with the restriction maps to open subsets.Proof. Recall that the K group of a Waldhausen category is the free abelian group generatedby the weak equivalence classes of objects of the category, modulo the relations induced by thecofibration sequences. The same is true for topological Waldhausen categories (cf. [Wei, 4.8.4.]). Inparticular since all the non-zero objects of the category Q A are isomorphic to each other by 1.24,hence K ( Q A ) = 0.In the case when A is a unital C ∗ -algebra, recall that Ext ( A ) [BDF77] is defined as the semi-group of unitary equivalence classes of unital injective representations of A to the Calkin algebra( B / K )( H ) (cf. [HR00, 2.7.1.]). When A is nuclear, then as a corollary of Voiculescu’s theorem we know that Ext ( A ) is in fact a group, and isomorphic to the first K-homology group K top ( A ). Lemma 4.2. Let A be a unital and nuclear C ∗ -algebra. Then K ( Q ′ A ) = K top ( A ) .Proof. Let ρ ′ i : A → ( B / K )( H i ) be non-zero objects in Q ′ A for i = 1 , 2, which are isomorphic,i.e. there exists isomorphisms T : ρ ′ → ρ ′ and S : ρ ′ → ρ ′ which are inverses to each other.By definition of a C ∗ -category, for a positive operator T ∗ T ∈ Q ′ A ( ρ ′ ), there exists an operator F ∈ Q ′ A ( ρ ′ ) so that F ∗ F = T ∗ T . Since S, T are invertible, then so are F and F S : ρ ′ → ρ ′ . Wehave ( F S ) ∗ ( F S ) = S ∗ F ∗ F S = S ∗ T ∗ T S = Id and (( F S )( F S ) ∗ ) F = ( F SS ∗ F ∗ ) F = F SS ∗ T ∗ T = F ST = F in the category Q ′ A . Hence F S is a unitary isomorphism in this category. Chooserepresentatives for S, F in the category D A . Because ρ ′ , ρ ′ are unital, then it means F S is also aFredholm operator, and in particular has closed image and finite dimensional kernel and cokernel.Hence there exists closed subspaces H ′ i ⊂ H i of finite codimension so that π F Sι : H ′ → H ′ isa isomorphism of Hilbert spaces, where ι i : H ′ i → H i is the inclusion and π i : H i → H ′ i is theprojection for i = 1 , 2. Since ker( F S ) = coker ( S ∗ F ∗ ) , ker( S ∗ F ∗ ) = coker ( F S ) then π F Sι is aunitary map of Hilbert spaces. Let ν ′ i = π i ρ ′ i ι i : A → ( B / K )( H ′ i ) for i = 1 , 2. Then we just showedthat ν ′ , ν ′ are unitarily equivalent. Since the difference between ρ ′ i , ν ′ i is a finite dimensional Hilbertspace, then they represent the same class in Ext ( A ) . This shows that Ext ( A ) → K ( Q ′ A ) isinjective. Surjectivity follows from the definition.By propositions 2.14 and 1.22, the maps of spectra induced by inclusion of subcategories K ( Q A ) → K (( D / C ) A ) and K ( Q ′ A ) → K (( D / C ) ′ A ) are both homotopy equivalences. By remark1.25 and proposition 2.14, the map K (( D / C ) A ) → K (( D / C ) ′ A ) induces an isomorphism on the i ’thK-groups for i ≥ A be a C ∗ -algebra, let ρ be an ample representation of A , and let R be a full subcategoryof Q A with two objects: the zero representation A → ρ . Then this is a C ∗ -category andalso a skeleton for the category Q A , as all ample representaions are isomorphic to each other. Butsince every short exact sequence in Q A splits, then by a result of Mitchener [Mit01], Ω k wS · Q A k ishomotopy equivalent to BGL ∞ ( Sk ( Q A )), where the latter is defined in [Mit01, 6.1.] and Sk ( Q A )denotes the skeleton of the additive category Q A . Hence the K-theory space of Q A is homotopyequivalent to BGL ( R ), which by definition is homotopy equivalent to BGL ∞ ( Q ( A )). Note that for a nuclear C ∗ -algebra A and C ∗ -algebra B with a C ∗ -ideal K , a ∗ -morphism A → B/K lifts to a completely positive map A → B (cf. [HR00, 3.3.6.]), and Stinespring’s theorem (cf. [HR00, 3.1.3.]) shows that eachcompletely positive map to B ( H ), can be written as V ∗ ρV , where V : H → H ′ is an isometry and ρ is a representationto B ( H ′ ). Then the restriction of ρ to the orthogonal complement of image of V induces a representation of A tothe Calkin algebra (cf. [HR00, 3.1.6.]). Let H ′′ i denote the orthogonal complement of H ′ i in H i , which is finite dimensional. Then the direct sum of thezero representation from A to ( B / K )( H ′′ i ) and ν ′ i is equal to ρ ′ i . i ≥ K − itop ( A ) ∼ = K topi ( Q ( A )) = π i ( BGL ( Q ( A ))) ∼ = π i ( BGL ∞ ( R )) ∼ = K topi ( Q A ) ∼ = K topi (( D / C ) A ) . (9)This means we have proved the first part of the theorem.Let A, B be unital C ∗ -algebras with ample representations ρ A : A → B ( H A ) and ρ B : B → B ( H B ). Let α : A → B be a unital map of C ∗ -algebras, then by Voiculescu’s theorem there existsan isometry V : H B → H A so that V ∗ ρ A ( a ) V − ρ B ( α ( a )) is compact for all a ∈ A . Note that V V ∗ ∈ B ( H A ) is a projection which commutes with the representation ρ A [HR00, 3.1.6.] modulocompact operators. Also note that V : H B → V V ∗ H A is an isomorphism of Hilbert spaces. Nowthe function Ad V ( T ) = V T V ∗ gives a map Ad V ( − ) : ( D / C ) B ( ρ B ) ∼ = Q ( B ) → Q ( A ) ∼ = ( D / C ) A ( ρ A ) , and hence induces a map on the K-homology groups which only depends on α , i.e. does not dependon the choices of ample representations ρ A , ρ B and the isometry V .Let T ∈ ( D / C ) B ( ρ B ) ∼ = Q ( B ) be a unitary element. The pull-back map of K-homology groupssends T to the unitary V ( T − Id H B ) V ∗ + Id H A = V T V ∗ ⊕ ( Id − V V ∗ ) ∈ ( D / C ) A ( ρ A ) ∼ = Q ( A ) . On the other hand, pulling back in the Paschke category is given by precomposing with the repre-sentation. Hence T ∈ ( D / C ) B ( ρ B ) is sent to T ∈ ( D / C ) A ( ρ B ◦ α ). These two procedures give twodifferent maps from the unitaries (or invertible elements) in ( D / C ) B ( ρ B ) to the topological spaceΩ k S · ( D / C ) A k .Let S = Id H B ⊕ ( V T V ∗ ⊕ ( Id H A − V V ∗ )) ∈ B ( H B ⊕ H A ). Consider the following two ”prisms”in wS · ( D / C ) A where the cofibration and the quotient maps on the left diagram are the trivial ones,but the cofibrations on the right diagram are given by (0 , V ) : H B → H B ⊕ H A and the quotientmaps are given by V + ( Id H A − V V ∗ ) : H B ⊕ H A → H A . By [HR00, 3.1.6.] these maps arepseudo-local. It is also easy to check that both diagrams below commute. ρ A ρ A ρ B ρ B ⊕ ρ A ρ A ρ B ρ B ⊕ ρ A ρ A ρ B ρ B ⊕ ρ A ρ B ρ B ⊕ ρ AV T V ∗ ⊕ ( Id − V V ∗ ) IdId S T S Since fat geometric realization of a point is the infinite dimensional ball which is contractible, thenin the fat geometric realization of wS · ( D / C ) A , the side of the prism ∆ top × ∆ top corresponding tothe identity map is contractible. Hence we get a homotopy from S to V T V ∗ ⊕ ( Id − V V ∗ ) inducedby the left diagram and also from S to T induced by the right diagram, by ”sliding” one side of theprism towards the other along the contractible side (corresponding to identity map). Therefore To be more precise, by the additivity theorem we know t, s + q : E ( D / C ) A → ( D / C ) A are homotopic, where E is temporarily denoting the category of cofibration sequences, and s, t, q refer to the first (source), second (target),and the third (quotient) object in the cofibration sequence. By applying this to the prism on the right we get that .In the context of Paschke categories restriction maps are defined similar to pull-back maps, i.e.by precomposing with the representation. To be more precise, let X be a locally compact andHausdorf topological space, and let U be an open subset. The inclusion j : U ֒ → X induces aninclusion j ∗ : C ( U ) ֒ → C ( X ) of C ∗ -algebras, given by extending functions by zero. Then therestriction map sends the object ρ : C ( X ) → B ( H ) to j ∗ ( ρ ) := ρ ◦ j ∗ : C ( U ) → B ( H ).We follow [RS13] to recall the process of defining the (wrong-way) restriction maps on theclassical topological K-homology. Let X, U, j, ρ : C ( X ) → B ( H ) be as before. If we extend ρ tothe Borel functions on X , then ρ ( U ) is a self-adjoint projection, where U is the characteristicfunction of the open subset U of X . Let H U be the image of this projection, and define therepresentation ρ U : C ( U ) → B ( H U ) by ρ U ( f ) = π U ρ ( j ∗ ( f )) ι U where ι U : H U → H is theinclusion, π U : H → H U is the projection, and j ∗ : C ( U ) → C ( X ) is extension by zero. Thelinear map B ( H ) → B ( H U ) defined by T π U T ι U maps D C ( X ) ( ρ ) to D C ( U ) ( ρ U ), and C C ( X ) ( ρ )to C C ( U ) ( ρ U ). Hence there is an induced map Q ρ ( C ( X )) → Q ρ U ( C ( U )), which induces therestriction map from the K-homology groups of X to the K-homology groups of U .But the representations j ∗ ρ, ρ U are naturally isomorphic; in fact the maps π U , ι U induce theisomorphisms. To show this, first note that π U , ι U commute with these representations since for f ∈ C ( U ), we have ρ U ( f ) π U = ρ U ( f ) ρ ( U ) = ρ ( j ∗ f ) = π U j ∗ ρ ( f ). Also ρ U ( π U ι U − Id H U ) and j ∗ ρ ( ι U π U − Id H ) are both zero. Therefore these two restriction functors are homotopic to each otheras a maps to the Ω k wS · ( D / C ) C ( U ) k , which means the two induced restriction maps on K-homologygroups are equal to each other. Definition 4.3. Let V C denote the topological category where the objects are finite dimensionalcomplex vector spaces, and morphisms are invertible linear maps. Let A be a nuclear C ∗ -algebra.Then there is a biexact functor V C × ( D / C ) ′ A → ( D / C ) ′ A induced by taking the tensor product ofthe corresponding Hilbert space with the finite dimensional vector space. This induces a map ofspectra ku ∧ K top (( D / C ) ′ A ) → K top (( D / C ) ′ A ) (10)where ku is the K-theory spectrum K top ( V C ), also known as the connective complex K-theoryspectrum .Let KU denote the (non-connective) complex K-theory spectrum .Since we have only defined the functor τ DX,g on relatively compact open subsets, we will need adescent argument to glue them together. So far we have defined a connective K-homology spectrumfor C ∗ -algebras. We need a non-connective K-homology spectrum to make the descent work. It is T + Id is homotopic to S , and by applying it on the prism on the left we get that ( V T V ∗ ⊕ ( Id − V V ∗ )) + Id ishomotopic to S , but since Id is contractible in the fat geometric realization, then we get that T is homotopic to V T V ∗ ⊕ ( Id − V V ∗ ). One may wonder why we did not simply say that V T V ∗ ⊕ ( Id − V V ∗ ) is direct sum of V T V ∗ and Id − V V ∗ ,and argue similar to above that the identity on the Hilbert space ( Id − V V ∗ ) H A corresponds to a contractible sideof a prism, and then ”slide” V T V ∗ ⊕ ( Id H A − V V ∗ ) directly onto V T V ∗ which is isomorphic to T in the category( D / C ) A , to obtain a homotopy. The reason is that the restriction of ρ A to ( Id − V V ∗ ) H A is only a representationup to compact operators, and the Hilbert space (1 − V V ∗ ) H A does not come with a representation, which means wecan not simply consider (1 − V V ∗ ) H A as an object in ( D / C ) A . Note that however, we can consider the Hilbert space V V ∗ H A together with the representation V ( ρ B α ) V ∗ . Lemma 4.4. By definition above, we can consider the smash product of spectra K top (( D / C ) ′ A ) ∧ ku KU . This has the same homotopy groups as the non-connective topological K-homology of A .Sketch Proof: Recall that for a ring spectrum X · and b ∈ X n , multiplication by b induces a map X · → Σ − n X · . Then we define X · [ b − ] to be the homotopy colimit of the telescope X · b −→ Σ − n X · Σ − n b −−−→ Σ − n X · Σ − n b −−−−→ Σ − n X · → . . . . Also, (stable) homotopy groups commute with this mapping telescope (cf. [EKM07, 5.1.14.]). Let β ∈ ku denote the bott element . Then it is well-known that KU is naturally homotopy equivalentto ku [ β − ] (cf. [Sna81]).Since K top (( D / C ) ′ A ) is a ku -module, then there is a natural weak equivalence K top (( D / C ) ′ A ) ∧ ku ku [ β − ] → K top (( D / C ) ′ A )[ β − ] (cf. [EKM07, 5.1.15.]). This is easy to see that the homotopy groupsof the latter is 2-periodic, as one could disregard the first n -temrs in the mapping telescope, andthat positive homotopy groups of K top (( D / C ) ′ A ) are 2-periodic. The fact that positive homotopygroups of K top (( D / C ) ′ A ) ∧ ku KU are isomorphic to the positive homotopy groups of K top (( D / C ) ′ A )follows from the (strongly converging) Atiyah-Hirzebruch spectral sequence (cf. [EKM07, 4.3.7.])and the fact that positive homotopy groups of ku and KU agree with each other. This finishes theproof. Proposition 4.5. Let A be a C ∗ -algebra, and let I ⊂ A be an ideal, so that the projection π : A → A/I has a completely positive section. Then K top (( D / C ) ′ A/I ) ∧ ku KU → K top (( D / C ) ′ A ) ∧ ku KU → K top (( D / C ) ′ I ) ∧ ku KU is a homotopy fiber sequence.Proof. It is easy to observe that the composition of the two maps above is null-homotopic. Henceit suffices to show that the homotopy groups of the sequence above induce a long exact sequenceof homotopy groups. This is a direct consequence of the six-term exact sequence of K-homologygroups [HR00, 5.3.10.], lemma 4.4 which says that the homotopy groups agree with the K-homologygroups and also theorem 9, which says that the pull-back maps of the Paschke category agree withthe classical pull-backs.We will give definition of descent with respect to hypercovers [DHI04, 4.2.] below. The definitionessentially states when a presheaf of spectra is in fact a sheaf (up to homotopy). A hypercover overa site X is a simplicial presheaf U · [Jar87, Sec 1.] with an augmentation U · → X , which satisfiescertain conditions. This can be thought of as a generalization of ˇCech covers , which have the form . . . Q j ,j ,j ( U j ∩ U j ∩ U j ) Q j ,j ( U j ∩ U j ) Q j ( U j ) X In fact, ˇCech covers are hypercovers of height zero. Also, for our purposes, homotopy limits referto limits in the homotopy category of spectra. Definition 4.6. [DHI04, 4.3.] Let X be an object in the site C (which can be thought of as atopological space). An object-wise fibrant simplicial presheaf F satisfies descent for a hypercover U · → X if the natural map from F ( X ) to the homotopy limit of the diagram Q i F ( U j ) Q j F ( U j ) Q j F ( U j ) . . . (11)34s a weak equivalence. Here the products range over the representable summands of each U n . If F is not object-wise fibrant, we say it satisfies descent if some object-wise fibrant replacement for F does.The fact that topological K-homology satisfies descent is essentially a result of the Atiyah-Hirzebruch spectral sequence (cf. [Bro73][AH61]). In fact, by [DI04, 4.3.], for a hypercover U · → X ,we have weak equivalences hocolim U · → | U · | → X , where the second map is induced by takinggeometric realization. Since taking smash product in the homotopy category of spectra preservescolimits, then by applying the smash product with an Ω-spectrum E · , the colimit of the diagram11 smashed with the spectrum E · is weakly equivalent to E · ∧ X . When the Ω-spectrum E · = KU is the (non-connective) topological K-theory spectrum, this proves descent. (Also see [AW14, 2.2.]for the case of twisted topological K-theory of CW-complexes.)Now we are ready to define the Riemann-Roch transformation over the relatively compact opensubsets of a complex manifold, which in turn induce the Riemann-Roch transformation over themanifold itself. Definition 4.7. Let X be a complex manifold, and let V be a relatively compact open subset of X . Define the functor τ X,V in the homotopy category of spectra as the composition below K alg ( P ( X )) y τ DV,g Defined in proposition 3.26. K alg ( Ch ′ ( D / C ) C ( V ) ) y τ HC ( V ) Defined in 3. K alg ( Bi ′ ( D / C ) C ( V ) ) y K alg ( Bi b ( D / C ) C ( V ) ) y τ G ( D / C ) C V ) Defined in 1.Ω K alg (( D / C ) C ( V ) ) y c Definition 2.5.Ω K top (( D / C ) C ( V ) ) y Ω K top (( D / C ) ′ C ( V ) ) ∧ ku KU y ∼ = By lemma 4.4. K top ( V )Note that except for the first one, all the maps above are functorially defined. Also, we showedin proposition 7 that in the homotopy category of spectra, τ D is compatible with restriction tofurther open subsets. Therefore for a hypercover V · so that all the open sets in V n are relativelycompact subsets of the open sets in V n − (and in particular, all are relatively compact in X ), thereis an induced map τ : K alg ( P ( X )) → holim K top ( V · ) in the homotopy category of spectra, wherethe latter is referring to the homotopy colimit of the diagram 11 for the hypercover V · → X . Sincetopological K-homology satisfies descent, then there is an induced map in the homotopy category35f spectra τ X : K alg ( P ( X )) → K top ( X ) . (12)By taking a finer cover if necessary, one can see that the map above is independent of the choiceof the hypercover V · . Proposition 4.8. The Riemann-Roch transformation defined above commutes with restriction toopen subsets. In other words, for a complex manifold X and an open subset U of X , the diagrambelow commutes in the homotopy category of spectra. K alg ( P ( X )) K alg ( P ( U )) K top ( X ) K top ( U ) τ X τ U Proof. Let V · → X be a hypercover so that all the open sets in V n are relatively compact in V n − .Choose a hypercover W · → U with the same condition as V · so that W n is finer than V n ∩ U , i.e. eachopen set in W n (which is a relatively compact open subset of U ) is contained in (the intersection of U with) some open set in V n . Hence for any relatively compact open set W jn in the hypercover W · there exists relatively compact open set V jn of the hypercover V · so that W jn ⊂ V jn ∩ U is relativelycompact. Hence by corollary 3.27 and by naturality of all the other maps in definition of τ , thediagram below commutes in the homotopy category of spectra. K alg ( P ( X )) K alg ( P ( U )) K alg ( Ch ′ ( D / C ) C ( V jn ) ) K alg ( Ch ′ ( D / C ) C ( W jn ) ) K top ( X ) K top ( V jn ) K top ( W jn ) τ DV jn τ X τ DWjn Hence there is a unique map from K top ( X ) to the holim K top ( W · ) that makes the diagram abovecommute (in the homotopy category of spectra), where the homotopy limit is taken on the diagram11 for the hypercover W · → U . But this homotopy limit is weakly equivalent to K top ( U ) becausetopological K-homology satisfies descent. This finishes the proof.We will investigate functoriality of τ with respect to proper morphisms of complex manifoldsin a future work. In the last subsection, let us emphasize on the case when the C ∗ -algebra A is unital, and how onecould define a pairing between the K-theory and K-homology of A using the Paschke category.When A is unital, we will define the Euler characteristic of an exact sequence in the Paschkecategory ( D / C ) A . Recall that if . . . → ρ i T ′ i −→ ρ i +1 T ′ i +1 −−−→ ρ i +2 → . . . 36s a chain complex in ( D / C ) A , and T i are representatives for T ′ i in D A , then the composition T i +1 ◦ T i may not be zero; it only has to be locally compact. If the representations are all unital, then T i +1 ◦ T i has to be compact. This is still not enough for us to be able to take the quotient of ker( T i +1 ) bythe image of T i , as the image may not even be a subset of the kernel. However, we can get aroundthis issue. Recall the following definition from [Seg70, Sec 1.] and the main result of [Tar07]. Definition 4.9. Let . . . → V i T i −→ V i +1 T i +1 −−−→ V i +2 → . . . be a complex of Hilbert spaces andbounded linear operators. Then it is called a Fredholm complex if all the T i ’s have closed imagesand the cohomology is finite dimensional at every step.Equivalently, we may define a complex of bounded operators between Hilbert spaces as before tobe a Fredholm complex if there exists bounded operators S i : V i +1 → V i so that T i − S i − + S i T i − Id V i is a compact operator for all i .If the complex is bounded, i.e. V i = 0 for all but finitely many values of i , then define its Eulercharacteristic by χ ( V · ) = P i ( − i H i ( V · )We can consider the Euler characteristic as a formal difference of two finite dimensional subspacesof ⊕ i V i . Proposition 4.10. Let . . . T ′ i − −−−→ V i T ′ i −→ V i +1 T ′ i +1 −−−→ . . . be a bounded above exact sequence in ( B / K ) . Then there are morphisms T i ∈ B ( V i , V i +1 ) so that T i is a representative for T ′ i , andalso T i +1 ◦ T i = 0 . Hence the new complex has a well-defined cohomology. Also the sequence . . . T i − −−−→ V i T i −→ V i +1 T i +1 −−−→ . . . is a Fredholm complex, and if the complex is both bounded aboveand below, then the Euler characteristic of the complex is independent of the choices of T i ’s.(In thesense that for the finite dimensional subspaces V + , V − , W of the Hilbert space H , we consider theformal differences V + − V − and V + ⊕ W − V − ⊕ W to be equivalent.) Let A be a unital C ∗ -algebra, and let . . . → ρ i T ′ i −→ ρ i +1 T ′ i +1 −−−→ ρ i +2 → . . . be a bounded exact sequence (i.e. there are only finitely many non-zero objects) in the Paschkecategory ( D / C ) A . Then by the argument proving proposition 1.22, we know there exists naturalchoices of unital representations ˆ ρ i which are isomorphic to ρ i . This induces a new exact sequence . . . → ˆ ρ i ˆ T i −→ ˆ ρ i +1 ˆ T i +1 −−−→ ˆ ρ i +2 → . . . where all the representations are unital and hence ˆ T i +1 ˆ T i is compact for all i , and this inducesan exact sequence in ( B / K ). Therefore by proposition 4.10, the exact sequence above has a well-defined euler characteristic. Note that this process can not be replicated for the Calkin-Paschkecategory, as the choice of the projection π ∈ B ( H ) corresponding to ρ ′ (1) may affect the index. Tosum it all up: Corollary 4.11. Let A be a unital C ∗ -algebra, and let ( ρ · , T · ) be an exact sequence in the Paschkecategory ( D / C ) A with finitely many non-zero objects. Then the procedure above defines the Eulercharacteristic of this complex. The Euler characteristic defined is additive with respect to exactsequences in the category Ch ′ ( D / C ) A . (This is not true for the category Ch b ( D / C ) A .) emark . When A is not unital, the argument in [Tar07] does not work anymore; as it relieson the fact that if Id H − ∆ ∈ K ( H ), then ∆ has a closed image. This is no longer the case if wereplace K ( H ) by locally compact operators.In a slightly different direction, let us define a natural pairing between projective modules andrepresentations of a C ∗ -algebra. Definition 4.13. Let R be a ring. Denote the exact category of finitely generated projective rightmodules on R by P r ( R ). When R is commutative, we drop the superscript r . Note that for any(right) projective R -module, there exists an integer n , and an inclusion ι : P → R n of (right) R -modules.We are interested in the particular case when R = A is a unital C ∗ -algebra. Let P rm ( A ) denotethe category of finitely generated projective right A -modules with an inner-product structure. Onecan consider the inclusion ι : P → A n of right A -modules to be norm preserving. Morphisms in P rm ( A ) are the (not necessarily norm-preserving) morphisms between the projective modules.Let X be a compact Hausdorf space, in particular a compact manifold. Then denote the exactcategory of topological (complex) vector bundles on X by P t ( X ). Recall the category P m ( X ) fromdefinition 3.8. Note that by the Serre-Swan theorem [Swa62, Thm 2.], P t ( X ) = P ( C ( X )), hence P tm ( X ) = P m ( C ( X )). Definition 4.14. Let A be a unital C ∗ -algebra and let P ∈ P mr ( A ). Let ρ be an object in ( D / C ) A .Then we define the representation P ⊗ ρ : A → B ( P ⊗ A H ), where we are considering H as a left A -module through the representation ρ .We follow [Ati70] to show that P ⊗ A H is in fact a Hilbert space, and hence the definition abovemakes sense.Since P is a finitely generated projective (right) module, there exists a norm preserving (right) A -module surjection π : A n → P , with a norm preserving (right) A -module section ι : P → A n .Without loss of generality, we can assume that ιπ is a self-adjoint projection on A ⊕ n . Now let ιπ ( e i ) = P j e j a ji for 1 ≤ i ≤ n, where e , . . . , e n are the standard basis for A n , and a ji ∈ A . Definethe linear operator ˆ π on H ⊕ n ∼ = A n ⊗ A H byˆ π ( e i ⊗ A h ) = X j e j ⊗ A ρ ( a ji ) h. It is easy to check that this is in fact a self-adjoint projection, and since ιπ is A -linear, then ˆ π alsocommutes with ρ ⊕ n , becauseˆ πρ ⊕ n ( a )( h , . . . , h n ) = X i ˆ π ( e i ⊗ A ρ ( a ) h i ) = X i,j ( e j ⊗ A ρ ( a ji ) ρ ( a ) h i ) = X i,j ( e j a ji ) ⊗ A ρ ( a ) h i = X i π ( e i ) ⊗ A ρ ( a ) h i = X i π ( e i .a ) ⊗ A h i = X i π ( a.e i ) ⊗ A h i = ρ ⊕ n ( a ) X i π ( e i ) ⊗ A h i = ρ ⊕ n ( a ) X i ˆ π ( e i ⊗ A h i ) = ρ ⊕ n ( a )ˆ π ( h , . . . , h n ) . Now let V ⊂ H ⊕ n be the image of ˆ π , and let ˆ ι : V → H ⊕ n be the inclusion, then consider thecomposition A ρ ⊕ n −−→ B ( H ⊕ n ) → B ( V ), where the last map sends T to ˆ πT ˆ ι (we are abusing thenotation and denoting the composition of ˆ π with the orthogonal projection H ⊕ n → V by ˆ π aswell.). It is easy to check that the compositions below are inverses to each other. V ˆ ι −→ H ⊕ n ∼ = A n ⊗ A H π ⊗ A Id −−−−→ P ⊗ A HV ˆ π ←− H ⊕ n ∼ = A n ⊗ A H ι ⊗ A Id ←−−−− P ⊗ A H A in D A , this induces structure of aHilbert space on P ⊗ A H . Proposition 4.15. Let A be a unital C ∗ -algebra. The tensor product introduced in definition 4.14,induces a biexact functor ∩ A : P rm ( A ) × ( D / C ) A → ( D / C ) A (13) which we will call the cap product . This induces a pairing on the level of K-theory spectra ∩ A : K alg ( P rm ( A )) ∧ K (( D / C ) A ) → K (( D / C ) A ) . (14) Proof. First we need to check functoriality. Let F : P → P be a morphism in P rm ( A ). Then wecan consider the morphism F ⊗ A Id : P ⊗ A ρ → P ⊗ A ρ in the category D A where p ⊗ A h F ( p ) ⊗ A h . Note that for a ∈ A , F ( p ) ⊗ A ρ ( a ) h = F ( p ) .a ⊗ A h = F ( p.a ) ⊗ A h . Hence this iswell defined and commutes with multiplication by A . It is clear that this process is functorial, i.e.( F ⊗ A Id ) ◦ ( F ⊗ A Id ) = F F ⊗ A Id in D A .Let T : ρ → ρ be a morphism in the Paschke category ( D / C ) A . Then we define Id ⊗ A T : P ⊗ A ρ → P ⊗ A ρ as follows. Let π : A n → P be a norm preserving surjective map of right A -modules, let ι : P → A n be the corresponding inclusion of right A -modules, and let ˆ π i ∈ B ( H ⊕ ni )be the projection corresponding to P ⊗ A H i for i = 1 , 2, and ˆ ι i be the inclusion of its image V i in the corresponding Hilbert space. Consider T ⊕ n : A n ⊗ A ρ ∼ = ρ ⊕ n → ρ ⊕ n ∼ = A n ⊗ A ρ , anddefine Id ⊗ A T = ˆ π T ⊕ n ˆ ι . Note that this is a pseudo-local operator, as both ˆ π , ˆ ι commute withthe representations and T is pseudo-local. Also, since T commutes with multiplication by a ji ∈ A modulo compact operators, then ˆ π T ⊕ n − T ⊕ n ˆ π is compact. This shows that the definition for Id ⊗ A T is independent of the choice of the projection π up to locally compact operators. This alsoshows that for morphisms T : ρ → ρ and T : ρ → ρ in the Paschke category, the compositions( Id ⊗ A T ) ◦ ( Id ⊗ A T ) = ˆ π T ⊕ n ˆ ι ˆ π T ⊕ n ι = ˆ π T ⊕ n T ⊕ n ˆ π ˆ ι are equal to each other modulo locallycompact operators. Hence this process is functorial. Remark . The map Id ⊗ A T : P ⊗ A ρ → P ⊗ A ρ is not well-defined in the category D A .Let ρ , ρ be two objects of ( D / C ) A , let T : ρ → ρ be a morphism, let P , P , be twoobjects of P rm ( A ), and let F : P → P be a morphism of right A -modules. Choose the normpreserving inclusions ι i : P i → A n i of right A -modules, so that there exist a map of right A -modules F ′ : A n → A n that makes the corresponding diagram commute. Then the square on theleft and the one on the right commutes in the category D A . Consider e i ⊗ A h ∈ A n ⊗ A H , wehave ( F ′ ⊗ A Id ) T ⊕ n ( e i ⊗ h ) = ( b ,i T ( h ) , . . . , b n ,i T ( h )), where b j,i is the j ’th term in F ′ ( e i ) ∈ A n ,and T ⊕ n ( F ′ ⊗ A Id )( e i ) = ( T ( b ,i h ) , . . . , T ( b n ,i h )). Since T is pseudo-local, then the square in thecenter also commutes in the Paschke category ( D / C ) A . Hence functoriality in two directions arecompatible with each other. P ⊗ ρ P ⊗ A ρ ρ ⊕ n ρ ⊕ n ρ ⊕ n ρ ⊕ n P ⊗ ρ P ⊗ A ρ ι F ⊗ A Id Id ⊗ A T F ⊗ A IdF ′ ⊗ A Id T ⊕ n F ′ ⊗ A Id ˆ π T ⊕ n ˆ π ˆ ι Id ⊗ A T 39t is easy to check that if T is invertible, then so is Id ⊗ A T , and if F is an isomorphism of A -modules, then F ⊗ A Id is a pseudo-local isomorphism of Hilbert spaces. This procedure is exactin both variables, because if T ◦ T = 0 then ( Id ⊗ A T ) ◦ ( Id ⊗ A T ) = 0 and similarly for F .Also, an exact sequence ( ρ · , T · ) in ( D / C ) A has a contracting homotopy S · , which translates into acontracting homotopy Id ⊗ A S · for the sequence ( P ⊗ A ρ · , Id ⊗ A T · ). Also a short exact sequence( P · , F · ) of projective modules splits, i.e. has a contracting homotopy which again gives a contractinghomotopy for the sequence ( P · ⊗ A ρ, F · ⊗ A Id ).Let F : P → P be an admissible monomorphism in P rm ( A ) and consider an exact sequence0 ρ ρ ρ T T S S with a choice of contracting homotopy in the Paschke category ( D / C ) A . Then there is a map P ⊗ A ρ ⊕ P ⊗ A ρ Id ⊗ A S ) ⊕ Id −−−−−−−−→ P ⊗ A ρ ⊕ P ⊗ A ρ → ( P ⊗ A ρ ) ∪ ( P ⊗ A ρ ) ( P ⊗ A ρ )which induces an isomorphism between the first object and the last object in the Paschke category( D / C ) A . The map P ⊗ A ρ ⊕ P ⊗ A ρ F ⊗ A S ,Id ⊗ A T ) t −−−−−−−−−−−−→ P ⊗ A ρ is an admissible monomorphism,whose cokernel is P ⊗ A ρ F ⊗ A T −−−−−→ P ⊗ ρ , where F : P → P is cokernel of F , and thecontracting homotopies are the trivial ones induced by contracting homotopies of ρ · and F · . Thisproves that ∩ A is biexact, hence by proposition 2.12 induces a map of K-theory spectra.Let f : A → B be a unital map between unital C ∗ -algebras. Recall there is an exact push-forward functor f ∗ : P r ( A ) → P r ( B ) and f ∗ : P rm ( A ) → P rm ( B ) defined by f ∗ ( P ) = P ⊗ A B . Thereis also a pull-back map f ∗ : ( D / C ) B → ( D / C ) A . One could ask about the relation between thepairing defined above and these structures. Proposition 4.17. The pairing defined in proposition 4.15 is natural in the sense that for a unitalmap f : A → B of unital C ∗ -algebras, the diagram below commutes up to homotopy K ( P rm ( A )) ∧ K (( D / C ) B ) K ( P rm ( B )) ∧ K (( D / C ) B ) K (( D / C ) B ) K (( D / C ) A ) . K ( P rm ( A )) ∧ K (( D / C ) A ) f ∗ × Id ∩ B f ∗ ∩ A Id × f ∗ Proof. Consider the diagram below 40 rm ( A ) × ( D / C ) B P rm ( B ) × ( D / C ) B ( D / C ) B ( D / C ) A . P rm ( A ) × ( D / C ) Af ∗ × Id ∩ B f ∗ ∩ A Id × f ∗ Let ρ : B → B ( H ) be a representation, and let P be an object in P r ( A ). We can consider H as aleft A -module through the representation f ∗ ρ : A → B → B ( H ). It is straightforward to check thatthe natural map of Hilbert spaces P ⊗ A H → ( P ⊗ A B ) ⊗ B H defined by p ⊗ A h ( p ⊗ A ⊗ B h iswell-defined, and has a two-sided inverse given by ( p ⊗ A b ) ⊗ B h p ⊗ A ρ ( f ( b )) h . This isomorphismis pseudo-local, hence induces a natural isomorphism between f ∗ (( P ⊗ A B ) ⊗ B ρ ) and P ⊗ A f ∗ ρ in the category D A . Hence the diagram above commutes up to natural isomorphisms. Remark . One can replace the Paschke category ( D / C ) A with the category Ch b ( D / C ) A (or C ( D / C ) A , C b ( D / C ) A , Ch ( D / C ) A ) in propositions 4.15 and 4.17 and the same result would stillhold. However, we can not necessarily replace Ch ′ ( D / C ) A , as morphisms in Ch ′ ( D / C ) A come from D A , but pairing a morphism with the identity on a projective module is only well-defined up tocompact operators.Fix an object ( ρ · , T · ) of Ch ′ ( D / C ) A , since for a morphism F : P → P in P rm ( A ), the morphism F ⊗ A Id : P ⊗ A ρ · → P ⊗ A ρ · is well-defined in D A , hence we obtain a functor P rm ( A ) → Ch ′ ( D / C ) A (15)which maps P to P ∩ A ( ρ · , T · ) = ( P ⊗ A ρ · , Id ⊗ A T · ). Definition 4.19. Let X be a compact complex manifold, let g be a hermitian metric on X , let X × C denote the trivial rank one bundle on X , and let E be a topological vector bundle on X .Then denote the map 15 obtained through pairing with τ DX,g ( X × C ) ∈ Ch ′ ( D / C ) C ( X ) defined in 4,by − ∩ τ D [ X ].We have to emphasize that for a non-holomorphic vector bundle, the Dolbeault complex is notwell-defined. Let X be a compact complex manifold, let E be a holomorphic vector bundle on X ,and let g, h be hermitian metrics on X, E , respectively. Recall from 3.3 that we have an exactsequence τ DX,g ( E, h ) in the Paschke category ( D / C ) C ( X ) corresponding to the Dolbeault complex. Proposition 4.20. Let X be a compact complex manifold, and let E be a holomorphic vector bundleon X . Choose hertmitian metrics g, h on X, E respectively. Then the chain complexes τ DX,g ( E, h ) and E ∩ τ D [ X ] are isomorphic to each other in the category Ch ′ ( D / C ) C ( X ) .Proof. Let π : X × C m → E be a smooth projection onto the bundle E , and let ι : E → X × C m be the inclusion. Denote the Dolbeault operator on the trivial bundle X × C k of rank k by D k = ¯ ∂ + ¯ ∂ ∗ , and let D E = ¯ ∂ E + ¯ ∂ ∗ E denote the Dolbeault operator on E . Note that by definition, Id ⊗ χ ( D ) ∈ B ( E ⊗ L ( X, ∧ , ∗ T ∗ X )) is defined as πχ ( D ) ⊕ m ι = πχ ( D m ) ι . By remark 3.24, πχ ( D m ) ι − πιχ ( D E ) is locally compact in the Paschke category ( D / C ) C ( X ) . Since πι = Id E , thisproves the assertion. 41 orollary 4.21. Let X be a compact complex manifold, and let E be a topological vector bundle.Then E ∩ τ DX,g is an exact sequence in the Paschke category ( D / C ) C ( X ) , and by corollary 4.11 has awell-defined Euler characteristic. By propositions 4.20 and 3.4, this concept of Euler characteristicis equal to the classical concept of the Euler characteristic of the Dolbeault complex when E is aholomorphic vector bundle. Appendix A Complex Manifolds Let us give the basics and the notation used for complex manifolds in here. A good source forreading more on the topic is [Wel07].Let X be a complex manifold and let E be a holomorphic vector bundle, then denote the sheaf ofholomorphic, real analytic, differentiable, and continuous sections of E by O ( E ) , C ω ( E ) , C ∞ ( E ) , C ( E ),respectively. Notice that each of the four sheaves just mentioned, is a subsheaf of the next ones.Also if X is only real analytic, then we can still consider the sheaves C ω ( E ) , C ∞ ( E ) , C ( E ), andsimilar statements can be repeated for differentiable or topological manifolds. Let S be one ofthe four sheaves above, then for an open subset U of X , denote the space of sections of E on U by S ( U, E ). In the case when E is the trivial line bundle X × C , then we will just denote S ( U )instead of S ( U, E ) and also denote the structure sheaf by S X .Let T ∗ X denote the cotangent bundle of the complex manifold X . Then the (almost) complexstructure of X induces the decomposition T ∗ X ⊗ R C = T ∗ ( X ) , ⊕ T ∗ ( X ) , , which in turn inducesthe Dolbeault operator ¯ ∂ : C ∞ ( ∧ p,q T ∗ X ) → C ∞ ( ∧ p,q +1 T ∗ X ), that vanishes on the holomorphicsections. Hence for a holomorphic vector bundle E , we get an induced differential operator¯ ∂ ⊗ C ∞ ( ∧ p,q T ∗ X ) ⊗ O O ( E ) → C ∞ ( ∧ p,q +1 T ∗ X ) ⊗ O O ( E ) , which is also known as the Dolbeault operator. But we have C ∞ ( ∧ p,q T ∗ X ) ⊗ O O ( E ) ∼ = O ( ∧ p,q T ∗ X ⊗ C E ). From now on, we will abbreviate the latter to A p,qX ( E ) (or just A p,q ( E ), if X is clear from thecontext.), and we denote the Dolbeault operator by ¯ ∂ E : A p,q ( E ) → A p,q +1 ( E ). We will also callthe following as the Dolbeault complex with coefficients in E :0 → A , X ( E ) ¯ ∂ E −−→ A , X ( E ) ¯ ∂ E −−→ A , X ( E ) ¯ ∂ E −−→ . . . ¯ ∂ E −−→ A ,nX ( E ) → n = dim C ( X ). Definition A.1. We follow [Wel07, 4.2.] to recall the definition of symbol of a differential operator.First let X be a differentiable manifold, and consider differentiable vector bundles E, F on X . Alinear operator D : C ∞ ( X, E ) → C ∞ ( X, F ) is a differential operator of order k , if no derivations oforder ≥ k + 1 appear in its local representation. We denote the vector space of all such operatorswith Diff k ( E, F ).Let T ′ X denote the cotangent bundle T ∗ X of X with the zero section deleted, and let π : T ′ X → X denote the projection. For k ∈ Z setSmbl k ( E, F ) := { σ ∈ Hom ( π ∗ E, π ∗ F ) : σ ( x, ρv ) = ρ k σ ( x, v ) , where ( x, v ) ∈ T ′ X, ρ > } . We now define the k -symbol of a differential operator as a linear map σ k : Diff k ( E, F ) → Smbl k ( E, F )by σ k ( D )( x, v ) e = D ( j k k ! ( g − g ( x )) k f )( x ) ∈ F x All the vector bundles and vector spaces we are considering in this section are over the complex numbers. Someof the arguments still hold over the real numbers as well. x, v ) ∈ T ′ X, e ∈ E x are given and g ∈ C ∞ ( X ) , f ∈ C ∞ ( X, E ) are chosen so that f ( x ) = e, dg x = v . We can see that we have a linear mapping σ k ( D )( x, v ) : E x → F x , and that thesymbol does not depend on the choices made.One can also define pseudo-differential operator of order k for k ∈ Z (which we will denote byPDiff k ), and their symbol, but since definitions are somewhat more technical, and will not be usedhere, we refer the interested reader to [Wel07, 4.3.].Symbols of (pseudo-) differential operators have the following important properties: σ k + m ( D D ) = σ m ( D ) σ k ( D ) when D ∈ PDiff k ( E , E ) , D ∈ PDiff m ( E , E ) σ k ( D ∗ ) = ( − k σ k ( D ) ∗ if D ∈ PDiff k ( E, F )where in here D ∗ ∈ PDiff k ( F, E ) is the formal adjoint of D [Wel07, 4.1.5.], and σ k ( D ) ∗ is theadjoint of the linear map σ k ( D )( x, v ) : E x → F x . Note that both D ∗ and σ ( D ∗ ) = σ ( D ) ∗ depend on the choice of metric on X and the bundles E, F . Definition A.2. [Wel07, 4.4.] Let E, F be differentiable vector bundles on the differentiablemanifold X and let D ∈ Diff k ( E, F ). Then we say that D is an elliptic differential operator if forall ( x, v ) ∈ T ′ X , the linear map σ k ( D )( x, v ) : E x → F x is an isomorphism. In particular both E, F have the same fiber dimension. The same can be defined for pseudo-differential operators.Let E , E , . . . , E m be a sequence of differentiable vector bundles on X and for some fixed k , let D i ∈ Diff k ( E i , E i +1 ) for all i = 0 , , . . . , m − 1. We say this is an elliptic complex if D i +1 ◦ D i = 0for all i , and also if the associated symbol sequence0 → π ∗ E σ k ( D ) −−−−→ π ∗ E σ k ( D ) −−−−→ π ∗ E σ k ( D ) −−−−→ . . . σ k ( D m − ) −−−−−−→ π ∗ E m → π : T ′ X → X is the projection. Remark A.3 . In the literature, elliptic complexes are usually defined for compact differentiablemanifolds, since Sobelov spaces over non-compact spaces don’t behave as well as they do on compactspaces (e.g. Rellich’s lemma works for Sobelov spaces over a fixed compact subset of the manifold.),which makes elliptic complexes over non-compact manifolds not as easy to work with. For example,the Hodge decomposition theorem (mentioned later in this section) is no longer true for non-compactcomplex manifolds. Example A.4. Let E be a holomorphic vector bundle on the complex manifold X . The Dolbeaultoperator ¯ ∂ E : A p,qX ( E ) → A p,q +1 X ( E ) is a differential operator of order 1, and for ( x, v ) ∈ T ′ X , and f ⊗ e ∈ ∧ p,q T ∗ x X ⊗ E , σ ( ¯ ∂ E )( x, v ) f ⊗ e = ( iv , ∧ f ) ⊗ e where in here v = v , + v , ∈ T ∗ x ( X ) , + T ∗ x ( X ) , . It is easy to check that the symbol sequenceis exact, and hence the Dolbeault complex 16 is an elliptic complex. Theorem A.5 (The Hodge decomposition) . Let X be a compact complex manifold, and let E be a holomorphic vector bundle on X . Choose hermitian metrics on X and on E and let ¯ ∂ ∗ i : A ,i +1 X ( E ) → A ,iX ( E ) be the formal adjoint of ¯ ∂ i : A ,iX ( E ) → A ,i +1 X ( E ) (with respect to themetrics chosen). Let ∆ i = ¯ ∂ i − ¯ ∂ ∗ i − + ¯ ∂ ∗ i ¯ ∂ i and let H ,i ( X, E ) = ker ∆ i ⊂ A ,iX ( E ) denote the harmonic (0 , i ) -forms. Then we have the orthogonal decomposition A ,iX ( E ) ∼ = H ,i ( X, E ) ⊕ im ( ¯ ∂ i − ) ⊕ im ( ¯ ∂ ∗ i ) , (17) and also there is an isomorphism H ,i ( X, E ) ∼ = H i ( X, E ) , where the latter, is the cohomology of X with coefficients in E . efinition A.6 (Hodge Star operator) . [Wel07, 5.1.] Let V be a (complex) vector space of di-mension n . Choose an inner product on V and then choose an orthonormal basis e , . . . , e n for V .Then define the Hodge ∗ -operator ⋆ : ∧ k V → ∧ n − k V defined by ⋆ ( e i ∧ . . . ∧ e i k ) = ± ( e j ∧ . . . ∧ e j n − k ), where { j , . . . , j n − k } is complement of { i , . . . , i k } in { , . . . , n } , and we assign the plus sign if { i , . . . , i k , j , . . . , j n − k } is an even permutation of { , . . . , n } , and assign the minus sign if it is an odd permutation.It is easy to extend ⋆ by linearity, and also to observe that ⋆ does not depend on the choice ofthe orthonormal basis, and depends only on the inner-product structure.Let X be a complex manifold of dimension n , and choose a hermitian metric g on X . Thensimilar to above, we can define the Hodge ⋆ -operator ⋆ : ∧ k T ∗ X → ∧ n − k T ∗ X and it is easy to see that there is an induced ⋆ -operator ⋆ : ∧ p,q T ∗ X → ∧ n − p,n − q T ∗ X. Let E be a holomorphic vector bundle on X , and choose a hermitian metric h on E . We canconsider the metric as a linear map h : E → E ∗ , where E ∗ is the dual vector bundle to E , andwe also have the dual linear map h ∗ : E ∗ → E , and these satisfy h ∗ h = Id E , hh ∗ = Id E ∗ . Let¯ ⋆ ( f ) := ⋆ ( ¯ f ) for a section f of ∧ ∗ , ∗ T ∗ X . Define¯ ⋆ E = ¯ ⋆ ⊗ h : ∧ p,q T ∗ X ⊗ E → ∧ n − q,n − p T ∗ X ⊗ E ∗ . Then one can show [Wel07, 5.2.4.a.] the following relation between the adjoint ¯ ∂ ∗ of the Dolbeaultoperator ¯ ∂ and the Hodge ⋆ -operator.¯ ∂ ∗ = − ¯ ⋆ E ∗ ¯ ∂ ¯ ⋆ E : A p,qX ( E ) → A p,q − X ( E ) . (18) Appendix B Functional Calculus Let us follow [HR00] to give a quick introduction to functional calculus. Definition B.1. Let T ∈ B ( H ). Then let C ∗ ( T ) temporarily denote the Banach subalgebra of B ( H ), generated by T , its adjoint T ∗ , and the identity operator.We say that the operator T is normal if T T ∗ = T ∗ T . If T is normal then C ∗ ( T ) is a commutativeBanach algebra.Let A be a unital Banach algebra. Then for a ∈ A , we defineSpectrum A ( a ) = { λ ∈ C : a − λ. A } . Proposition B.2 (Spectral Theorem) . [HR00, 1.1.11.] Let T ∈ B ( H ) be a bounded normaloperator acting on the Hilbert space H , then the map α α ( T ) is a homomorphism from dual of C ∗ ( T ) onto Spectrum B ( H ) ( T ) , and the induced Gelfand transform C ∗ ( T ) → C ( Spectrum B ( H ) ( T )) is an isometric ∗ -isomorphism. Definition B.3. Let T ∈ B ( H ) be a bounded normal operator acting on the Hilbert space H ,and let f ∈ C (Spectrum B ( H ) ( T )). Denote the corresponding element in C ∗ ( T ) by f ( T ). The ∗ -homomorphism (inverse of the Gelfand transform) C (Spectrum B ( H ) ( T )) → B ( H ) defined by f f ( T ) is called functional calculus for T . 44 efinition B.4. [HR00, 1.8.] Let T be an unbounded operator, defined over a dense subset of theHilbert space H . Then we say T is symmetric if for each x, y ∈ H which are in domain of T , wehave h T x, y i = h x, T y i .We say T is essentially self-adjoint if domain of T is a subset of domain of T ∗ , and for any x indomain of T , T x = T ∗ x , and also x is in domain of T ∗ if there is a sequence of points { x i } ∞ i =1 indomain of T so that x i ’s converge to x and k T ( x i ) k remains bounded.Note that the first two conditions are equivalent to T being symmetric. In other words, everyessentially self-adjoint unbounded operator is symmetric. Lemma B.5. [HR00, 10.2.6.] Every symmetric differential operator on a compact manifold isessentially self-adjoint. More generally, every compactly supported symmetric differential operatoron a (non-compact) manifold is essentially self-adjoint. References [AA67] Michael Francis Atiyah and DW Anderson. K-theory , volume 2. WA Benjamin NewYork, 1967.[AA95] John Frank Adams and John Frank Adams. Stable homotopy and generalised homology .University of Chicago press, 1995.[AH61] Michael F Atiyah and Friedrich Hirzebruch. Vector bundles and homogeneous spaces. In Proc. Sympos. Pure Math , volume 3, pages 7–38, 1961.[Ati70] Michael F Atiyah. Global theory of elliptic operators. In Proc. Internat. Conf. onFunctional Analysis and Related Topics (Tokyo, 1969) , pages 21–30, 1970.[AW14] Benjamin Antieau and Ben Williams. The period-index problem for twisted topologicalk–theory. Geometry & Topology , 18(2):1115–1148, 2014.[BDF77] Lawrence G Brown, Ronald George Douglas, and Peter A Fillmore. Extensions of c*-algebras and k-homology. Annals of Mathematics , pages 265–324, 1977.[Bro73] Kenneth S Brown. Abstract homotopy theory and generalized sheaf cohomology. Trans-actions of the American Mathematical Society , 186:419–458, 1973.[DHI04] Daniel Dugger, Sharon Hollander, and Daniel C Isaksen. Hypercovers and simplicialpresheaves. In Mathematical Proceedings of the Cambridge Philosophical Society , volume136, pages 9–51. Cambridge University Press, 2004.[DI04] Daniel Dugger and Daniel C Isaksen. Topological hypercovers and 1-realizations. Math-ematische Zeitschrift , 246(4):667–689, 2004.[EKM07] Anthony D Elmendorf, Igor Kriz, and Michael A Mandell. Rings, modules, and algebrasin stable homotopy theory . Number 47. American Mathematical Soc., 2007.[Gil81] Henri Gillet. Riemann-roch theorems for higher algebraic k-theory. Advances in Mathe-matics , 40(3):203–289, 1981.[GLR85] Paul Ghez, Ricardo Lima, and John Roberts. W-categories. Pacific Journal of Mathe-matics , 120(1):79–109, 1985. 45Gra92] Daniel R Grayson. Adams operations on higherk-theory. K-theory , 6(2):97–111, 1992.[Gra12] Daniel Grayson. Algebraic -theory via binary complexes. Journal of the American Math-ematical Society , 25(4):1149–1167, 2012.[Hig95] Nigel Higson. C*-algebra extension theory and duality. Journal of Functional Analysis ,129(2):349–363, 1995.[HR00] Nigel Higson and John Roe. Analytic K-homology . OUP Oxford, 2000.[Jar87] John Frederick Jardine. Simplical presheaves. Journal of Pure and Applied Algebra ,47(1):35–87, 1987.[K + 00] Tamaz Kandelaki et al. kk -theory as the k -theory of c* -categories. Homology, Homotopyand Applications , 2(1):127–145, 2000.[Kar68] Max Karoubi. Algebres de Clifford et K-th´eorie . PhD thesis, Gauthier-Villars, 1968.[Kar08] Max Karoubi. K-theory: An introduction , volume 226. Springer Science & BusinessMedia, 2008.[Kas80] Gennadii Georgievich Kasparov. The operator k-functor and extensions of cˆ*-algebras. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya , 44(3):571–636, 1980.[Lax07] Peter D. Lax. Linear algebra and its applications . Pure and Applied Mathematics (Hobo-ken). Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, second edition, 2007.[Lev87] Roni N Levy. The riemann-roch theorem for complex spaces. Acta Mathematica ,158(1):149–188, 1987.[Lev08] Roni N Levy. Riemann-roch theorem for higher bivariant k-functors. In Annales del’institut Fourier , volume 58, pages 571–602. Chartres: L’Institut, 1950-, 2008.[Mit01] Paul D Mitchener. Symmetric Waldhausen K-theory spectra of topological categories .Institut for Matematik og Datalogi, SDU-Odense Universitet, 2001.[Mit02] Paul D Mitchener. C*-categories. Proceedings of the London Mathematical Society ,84(2):375–404, 2002.[Pas81] William Paschke. K-theory for commutants in the calkin algebra. Pacific Journal ofMathematics , 95(2):427–434, 1981.[Qui73] Daniel Quillen. Higher algebraic k-theory: I. In Higher K-theories , pages 85–147.Springer, 1973.[RS13] John Roe and Paul Siegel. Sheaf theory and paschke duality. Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology , 12(02):213–234, 2013.[Seg68] Graeme Segal. Classifying spaces and spectral sequences. Publications Math´ematiquesde l’IH ´ES , 34:105–112, 1968.[Seg70] Graeme Segal. Fredholm complexes. The Quarterly Journal of Mathematics , 21(4):385–402, 1970.[Seg74] Graeme Segal. Categories and cohomology theories. Topology , 13(3):293–312, 1974.46Sna81] Victor Snaith. Localized stable homotopy of some classifying spaces. In MathematicalProceedings of the Cambridge Philosophical Society , volume 89, pages 325–330. Cam-bridge University Press, 1981.[Swa62] Richard G Swan. Vector bundles and projective modules. Transactions of the AmericanMathematical Society , 105(2):264–277, 1962.[Tar07] Nikolai Nikolaevich Tarkhanov. Euler characteristic of fredholm quasicomplexes. Func-tional Analysis and Its Applications , 41(4):318–322, 2007.[TT90] Robert W Thomason and Thomas Trobaugh. Higher algebraic k-theory of schemes andof derived categories. In The Grothendieck Festschrift , pages 247–435. Springer, 1990.[Voi76] Dan Voiculescu. A non-commutative weyl-von neumann theorem. Rev. Roum. Math.Pures et Appl. , 21:97–113, 1976.[Wal85] Friedhelm Waldhausen. Algebraic k-theory of spaces. Algebraic and geometric topology(New Brunswick, NJ, 1983) , 1126:318–419, 1985.[Wei] Charles A Weibel. The K-book , volume 145.[Wel07] Raymond O Wells.