Thomsen's D-theory and the K-theoretic Kronecker pairing
aa r X i v : . [ m a t h . OA ] S e p Thomsen’s D-theory and the K-theoreticKronecker pairing
Benedikt HungerSeptember 8, 2020
We calculate Thomsen’s D-theory groups D (Σ , B ), where Σ = C ( R ). Fur-thermore, we relate the pairing K ( A ) × E ( A, B ) → K ( B ) to a similarpairing which is defined using D-theory. Connes and Higson [CH90] introduced the so-called E-theory groups E ( A, B ) as themorphism sets in an additive category E whose objects are the separable C*-algebras.There is a natural functor from the category of C*-algebras to the category E whichhas the universal property that every stable, homotopy-invariant, and half-exact functorfrom the category of C*-algebras to an additive category factors through E . In particular,E-theory is closely related to Kasparov’s KK-theory [Kas80]. E-theory was successfullyused by Higson and Kasparov [Hig00; HK97] to prove special cases of the Baum–Connesconjecture.In [Tho03], Thomsen introduced D-theory, a discrete variant of Connes’s and Hig-son’s E-theory. Like E-theory, D-theory is a bifunctor from the category of separableC*”=algebras to the category of abelian groups, and there exist products D ( A, B ) × D ( B, C ) → D ( A, C ) ,D ( A, B ) × E ( B, C ) → D ( A, C ) ,E ( A, B ) × D ( B, C ) → D ( A, C ) , relating E-theory and D-theory.It is a well-known calculation that the E-theory groups satisfy E ( C , B ) ∼ = K ( B ) forall C*-algebras B . However, there is, up to now, no concrete calculation of D-theorygroups at all. Of course, one might hope to calculate D ( C , B ) in a similar way. SinceD-theory (like E-theory) supports a suspension isomorphism Σ : D ( A, B ) ∼ = → D (Σ A, Σ B )for all separable C*-algebras A and B (where Σ A = C ( R ) ⊗ A is the suspension of A ),it suffices to calculate the groups D (Σ , B ) for Σ = Σ C = C ( R ).1hus, the aim of the first part of this paper will be to prove the existence of a naturalisomorphism D (Σ , B ) ∼ = Q n ∈ N K ( B ) L n ∈ N K ( B ) (1)where Σ = C ( R ) is the suspension C*-algebra. The proof of (1) is similar, in structure,to the proof of the natural isomorphism E ( C , B ) ∼ = K ( B ). However, the technicaldetails are slightly more complicated and involve the use of the concrete form of Cuntz’speriodicity map [Cun84]. Product-modulo-sum quotients as in the right hand of (1) haveappeared before in various contexts, for example in the work of Blackadar and Kirchberg[BK97] on inductive limits of C*-algebras, of Hanke and Schick [HS06; HS07; HS08] orof Carri´on and Dadarlat [CD18] on the maximal Baum–Connes assembly map.There exists a natural pairing K ( A ) × E ( A, B ) ∼ = E ( C , A ) × E ( A, B ) → E ( C , B ) ∼ = K ( B ), defined using the E-theory product [CH90]. By (1) there is also a pairing Q n ∈ N K ( A ) L n ∈ N K ( A ) × E ( A, B ) ∼ = D (Σ , A ) × E ( A, B ) → D (Σ , B ) ∼ = Q n ∈ N K ( B ) L n ∈ N K ( B )defined using Thomsen’s product. In the second part of this paper, we will showthat this pairing agrees with the pairing Q n ∈ N K ( A ) / L n ∈ N K ( A ) × E ( A, B ) → Q n ∈ N K ( B ) / L n ∈ N K ( B ) induced by the E-theory product. This shows that it is pos-sible to investigate the asymptotic behaviour of the pairing K ( A ) × E ( A, B ) → K ( B )using Thomsen’s D-theory product.This paper is based on part of the author’s doctoral dissertation at the Universit¨atAugsburg. The author would like to thank his advisor Bernhard Hanke for his supportand advice, and his helpful remarks on first version of this paper. The dissertationproject was supported by a scholarship of the Studienstiftung des deutschen Volkes andby the TopMath program of the
Elitenetzwerk Bayern . We will define the D-theory groups, and in particular Thomsen’s products, in a way whichdiffers slightly from [Tho03], and which is inspired by the definition of the correspondingobjects for E-theory in [GHT00]. We will also review the definition of E-theory, as wewill need it later on.If X is a locally compact Hausdorff space and B is a C*-algebra, we denote by C b ( X ; B )the C*-algebra of bounded continuous B -valued continuous functions on X , and by C ( X ; B ) ⊂ C b ( X ; B ) the ideal of functions vanishing at infinity. We are particularlyinterested in the special case X = P = [0 , ∞ ), where we abbreviate T B = C b ( P ; B ) and T B = C ( P ; B ). Similarly, we write T δ B = C b ( N ; B ), T δ, B = C ( N ; B ). Definition 2.1.
The discrete asymptotic algebra [Tho03] over B is the C*-algebra A δ B = T δ B/ T δ, B , and the asymptotic algebra [GHT00] over B is A B = T B/ T B .2very *-homomorphism f : A → B induces, by postcomposition, *-homomorphisms C b ( X ; A ) → C b ( X ; B ) which restrict to *-homomorphisms C ( X ; A ) → C ( X ; B ). Inparticular, f also induces *-homomorphisms A δ A → A δ B and A A → A B . These defini-tions turn A and A δ into functors, which are easily seen to be exact. Definition 2.2. An asymptotic homomorphism between C*-algebras A and B is a *-homomorphism A → A B . Similarly, a discrete asymptotic homomorphism is a *-homo-morphism A → A δ B .The E-theory and D-theory groups are defined as equivalence classes of asymptotic ho-momorphisms and discrete asymptotic homomorphisms, respectively, modulo an appro-priate type of homotopy. In fact, it turns out that the usual notion of homotopy of*-homomorphisms is too restrictive for asymptotic homomorphisms, and that the follow-ing provides a useful notion of homotopy between (discrete) asymptotic homomorphisms. Definition 2.3. An asymptotic homotopy is a *-homomorphism H : A → A IB , where IB = C ([0 , , B ) is the C*-algebra of continuous B -valued functions on the unit interval[0 , A ev ◦ H and A ev ◦ H from A → B are thencalled asymptotically homotopic . Here ev τ : IB → B denotes the evaluation at τ ∈ [0 , discrete asymptotic homotopy is a *-homomorphism H : A → A δ IB , andagain we call A δ ev ◦ H and A δ ev ◦ H asymptotically homotopic .One can show [GHT00, Proposition 2.3] that asymptotic homotopy defines an equiva-lence relation on the sets of (discrete) asymptotic homomorphisms from A to B . Definition 2.4.
We denote the set of asymptotic homotopy classes of asymptotic ho-momorphisms from A to B by [[ A, B ]], and the set of asymptotic homotopy classes ofdiscrete asymptotic homomorphisms from A to B by [[ A, B ]] δ . Definition 2.5. If A and B are separable C*-algebras, then we define E ( A, B ) = [[Σ A ⊗ K , Σ B ⊗ K ]]and D ( A, B ) = [[Σ A ⊗ K , Σ B ⊗ K ]] δ , where Σ A = { φ ∈ IA : φ (0) = φ (1) = 0 } ∼ = C ( R ) ⊗ A is the suspension of A , and where K = K ( ℓ ) is the C*-algebra of compact operators on a separable Hilbert space.Note that the definitions of the E-theory and D-theory groups directly imply that thereare stability isomorphisms E ( A, B ) ∼ = E ( A ⊗K , B ) ∼ = E ( A, B ⊗K ) and analogous stabilityisomorphisms for the D-theory groups.The double suspension which appears in the definition of D ( A, B ) in front of the C*-algebra B has its origin in the following alternative viewpoint towards the discrete asymp-totic algebra: The inclusion N ⊂ P = [0 , ∞ ) induces restriction maps T B → T δ B and The separability assumption here is in order to make sure that we can define the products later on. B → T δ, B for all C*-algebras B . These maps are clearly natural, so the induce a nat-ural transformation A → A δ , which is clearly surjective. Thus, we may define anotherfunctor by considering the kernel of this natural transformation. Definition 2.6.
The sequentially trivial asymptotic algebra [Tho03] over B is the C*-algebra A B = ker( A B → A δ B ). A sequentially trivial asymptotic homomorphism is a*-homomorphism A → A B , and a sequentially trivial asymptotic homotopy is a *-homo-morphism A → A IB as before. We denote by [[ A, B ]] the set of asymptotic homotopyclasses of sequentially trivial asymptotic homomorphisms.There is a natural equivalence between the functors A δ ◦ Σ and A [cf. Tho03, Lemma5.4]. Indeed, we may define maps η B : T δ Σ B → T B by η B ( φ )( t ) = φ ( ⌊ t ⌋ )( t − ⌊ t ⌋ ) . Then η B ( φ ) ∈ T B whenever φ ∈ T δ, Σ B , so there is an induced *-homomorphism η B : A δ Σ B → A B . Lemma 2.7.
The *-homomorphism η B : A δ Σ B → A B is injective, and its image equals A B . Thus, η B : A δ Σ B → A B is a natural *-isomorphism.Proof. It suffices to prove that the rightmost column in the diagram0 0 00 T δ, Σ B T δ Σ B A δ Σ B T B T B A B T δ, B T δ B A δ B
00 0 0 η B η B is exact. Since the rows of the diagram are exact by definition, it suffices, by the NineLemma, to prove that the two left columns are exact, which is straightforward.Hence, there are natural bijections [[ A, Σ B ]] δ → [[ A, B ]] , and we may equally well define D ( A, B ) by D ( A, B ) = [[Σ A ⊗ K , Σ B ⊗ K ]] . This description is most useful for constructing the products relating D-theory and E-theory. These products were introduced by Connes and Higson [CH90] for E-theory, and4y Thomsen [Tho03] for D-theory. The E-theory products were defined in a slightlydifferent way by Guentner, Higson and Trout [GHT00]. We will give definitions of theproducts which combine the methods of Thomsen and of Guentner, Higson and Trout.Indeed, we will use the following statement from [GHT00] to simplify the definition givenin [Tho03] significantly.
Lemma 2.8 ([GHT00, Claim 2.18]) . Consider a C*-subalgebra E ⊂ T B which isseparable. Then there exists a piecewise linear invertible continuous function r : P → P with lim t →∞ r ( t ) = ∞ , such that lim sup t →∞ sup r ≥ r ( t ) k F ( t )( r ) k ≤ k [ π ◦ F ] k A B for all F ∈ E , where π : T B → A B is the projection. Any such function r will becalled an admissible reparametrization for E ⊂ T B . Now the key idea in the definition of the products is the following: Suppose that f : A →A B and g : B → A C are asymptotic homomorphisms. Then we may consider A g ◦ f : A → A C . Thus, if we had a natural map Φ : A C → A C then we could use thismap to define the product g • f = Φ ◦ A g ◦ f : A → A C .Using Lemma 2.8, this idea can be made rigorous provided that A is separable. Indeed,suppose that E ⊂ A C is a separable C*-subalgebra. Let as C : T C → A C be thenatural projection. Then we can choose a separable C*-subalgebra E ⊂ T C with E ⊂ as C ( E ), and an admissible reparametrization r : P → P for E . We define *-ho-momorphisms Φ , ˆΦ : E → A C byΦ([ π ◦ F ]) = [ t F ( t )( r ( t ))] , ˆΦ([ π ◦ F ]) = [ t F ( r − ( t ))( t )]for F ∈ E with as C ( F ) = [ π ◦ F ] ∈ E . Lemma 2.8 makes sure that Φ and ˆΦ arewell-defined *-homomorphisms which, however, certainly do depend on the choices of E and r .Now suppose that f : A → A B and g : B → A C are asymptotic homomorphisms. If A is separable, then also A g ( f ( A )) ⊂ A C is separable. Choose a separable C*-subalgebra E ⊂ T C with A g ( f ( A )) ⊂ as C ( E ), and fix an admissible reparametrization r : P → P for E . Let Φ : A g ( f ( A )) → A C be a as constructed above. Then we define theasymptotic composition of f and g to be g • f = Φ ◦ A g ◦ f : A → A C . This constructiongives a product [[ A, B ]] × [[ B, C ]] → [[ A, C ]], ([ f ] , [ g ]) [ g • f ], and hence a product E ( A, B ) × E ( B, C ) → E ( A, C ) if A is separable. It turns out that the choices whichwent into the definition of Φ do not change this product, and that one could equally wellreplace Φ by ˆΦ.Now the situation is very similar when one considers sequentially trivial asymptotichomomorphisms: If f : A → A B is sequentially trivial and g : B → A C is as above, Hence π ◦ F : P → A B represents an element of A B . E ⊂ T C with A g ( f ( A )) ⊂ as C ( E ) in such a way that every F ∈ E satisfies F ( n )( t ) = 0 for all n ∈ N and t ∈ P . Then Φ ◦ A g ◦ f : A → A C issequentially trivial as well. Similarly, if f : A → A B is arbitrary and g : B → A C issequentially trivial then we may choose E such that F ∈ E satisfies F ( t )( n ) = 0 forall n ∈ N and t ∈ P . In this case, we may put g • f = ˆΦ ◦ A g ◦ f . Thus, we havedefined products [[ A, B ]] × [[ B, C ]] → [[ A, C ]] and [[ A, B ]] × [[ B, C ]] → [[ A, C ]] . Usingeither of the natural maps [[ A, B ]] → [[ A, B ]] or [[
B, C ]] → [[ B, C ]], we also obtain aproduct [[
A, B ]] × [[ B, C ]] → [[ A, C ]] . All of these products are compatible with themaps [[ · , · ]] → [[ · , · ]], and they satisfy all possible kinds of associativity laws [cf. Tho03,Section 3].There are two natural ways to define a group structure on the sets E ( A, B ) and D ( A, B ).The first one, which is employed in [GHT00] and [Tho03], goes as follows: Choose aunitary isomorphism V : ℓ → ℓ ⊕ ℓ . Then conjugation with V induces a C*-algebraisomorphism K ( ℓ ⊕ ℓ ) → K ( ℓ ), which is independent of the choice of V up to homotopy.We may therefore define a product on the set [[ A, Σ B ⊗ K ]] by the composition[[ A, Σ B ⊗ K ]] × [[ A, Σ B ⊗ K ]] ∼ = [[ A, Σ B ⊗ ( K ⊕ K )]] → [[ A, Σ B ⊗ K ( ℓ ⊕ ℓ )]] ∼ = [[ A, Σ B ⊗ K ]] . Another way would be to employ the natural map Σ ⊕ Σ → Σ given by concatenation,where Σ ∼ = C (0 , A, Σ B ⊗ K ]] agreeand define an abelian group structure on [[ A, Σ B ⊗ K ]]. In an entirely analogous fashion,also [[ A, Σ B ⊗ K ]] and [[ A, Σ B ⊗ K ]] δ (and hence in particular D ( A, B ) and E ( A, B )) areabelian groups.One can show that the asymptotic products described above define group homomor-phisms D ( A, B ) × D ( B, C ) → D ( A, C ), D ( A, B ) × E ( B, C ) → D ( A, C ), E ( A, B ) × D ( B, C ) → D ( A, C ), and E ( A, B ) × E ( B, C ) → E ( A, C ). D (Σ , B ) In this section, we will give a calculation of the group D (Σ , B ), for any separable C*-algebra B . This calculation bears some similarities with the well-known natural isomor-phism E ( C , B ) ∼ = K ( B ), which we will review first.Let B be an arbitrary C*-algebra, and consider a unitary u ∈ M n ( B + ) with u − ∈ M n ( B ). As usual, we denote the set of such unitaries by U + n ( B ) ⊂ M n ( B + ). Inparticular, u represents an element [ u ] ∈ K ( B ). We identify the C*-algebra Σ = C (0 , φ ∈ C ( S ) with φ (1) = 0. Then there exists a unique*-homomorphism g u : Σ → M n ( B ) ⊂ B ⊗ K with g u ( ω ) = u − ω : S → C isgiven by ω ( z ) = z −
1. We define a map g B : K ( B ) → [[Σ , B ⊗ K ]] Here B + is the unitization of the C*-algebra B .
6y [ u ] [ κ B ⊗K ◦ g u ] where κ B ⊗K : B ⊗ K → A ( B ⊗ K ) is given by κ B ⊗K ( x ) = [ t x ].The following statement is well-known: Proposition 3.1 ([Ros82, Theorem 4.1] and [GHT00, Proposition 2.19]) . For everyC*-algebra B , the map g Σ B : K (Σ B ) → [[Σ , Σ B ⊗ K ]] is an isomorphism of groups. One can show easily that the inclusion Σ → Σ ⊗ K induces an isomorphism E ( C , B ) =[[Σ ⊗ K , Σ B ⊗ K ]] → [[Σ , Σ B ⊗ K ]]. Together with Proposition 3.1, this establishes theisomorphism K ( B ) ∼ = K (Σ B ) ∼ = E ( C , B ) for arbitrary C*-algebras B .Now let us turn to the calculation of D (Σ , B ), which has not appeared in the literatureso far. Let again B be a C*-algebra, and let ( u n ) n ∈ N be a sequence in S k ∈ N U + k ( B ), sothat each u n represents an element of K ( B ). The map φ : N → B ⊗ K , which is definedby φ ( n ) = u n −
1, determines an element [ φ ] ∈ A δ ( B ⊗ K ) such that [ φ ] + 1 ∈ A δ ( B ⊗ K ) + is unitary. Hence there exists a unique discrete asymptotic homomorphism ˜ g ( u n ) : Σ →A δ ( B ⊗ K ) such that ˜ g ( u n ) ( ω ) = [ φ ] = [ n u n − g Bδ : Y n ∈ N K ( B ) → [[Σ , B ⊗ K ]] δ , ([ u n ]) n ∈ N [˜ g ( u n ) ] , is well-defined: Indeed if we are given continuous paths ( u τn ) τ ∈ [0 , in U + k ( n ) ( B ) then thesame construction as above yields a discrete asymptotic homotopy H : Σ → A δ I ( B ⊗ K )with H ( ω ) = [ n ( τ u τn − H is a discrete asymptotic homotopy connect-ing ˜ g ( u n ) and ˜ g ( u n ) . The key step in calculating D (Σ , B ) is the following analogue ofProposition 3.1. Proposition 3.2.
For every C*-algebra B , the map g Σ Bδ : Q n ∈ N K (Σ B ) → [[Σ , Σ B ⊗K ]] δ is a surjective group homomorphism with ker g Σ Bδ = M n ∈ N K (Σ B ) , where L n ∈ N K (Σ B ) ⊂ Q n ∈ N K (Σ B ) is the subgroup consisting of all sequences ([ u n ]) n ∈ N which vanish eventually.Proof. We begin by proving that g Σ Bδ is surjective. Thus, we consider an arbitraryelement [ h ] ∈ [[Σ , Σ B ⊗ K ]] δ which is represented by a discrete asymptotic homomorphism h : Σ → A δ (Σ B ⊗ K ). We may write h ( ω ) = [ G ] for a map G : N → Σ B ⊗ K . We mayreplace each G ( n ) by an element of S k ∈ N M k (Σ B ) ⊂ Σ B ⊗ K which is n − -close to G ( n ),without altering [ G ] ∈ A δ (Σ B ⊗ K ). Thus, we may assume that G ( n ) ∈ S k ∈ N M k (Σ B )for all n ∈ N .We will show next that there exists a map U : N → Σ B ⊗ K such that U ( n ) ∈ S k ∈ N U + k (Σ B ) for all n ∈ N , and such that [ G ] = [ U − ∈ A δ (Σ B ⊗ K ). Indeed,[ G + 1] ∈ A δ (Σ B ⊗ K ) must be unitary, so thatlim n →∞ ( G ( n ) + 1) ∗ ( G ( n ) + 1) = 1 .
7e put F ( n ) = G ( n ) + 1. Thus, F ( n ) ∗ F ( n ) is invertible if n is sufficiently large.Without loss of generality, F ( n ) ∗ F ( n ) is invertible for all n ∈ N . Now we put U ( n ) = F ( n )( F ( n ) ∗ F ( n )) − / . It is straightforward to see that indeed [ G ] = [ F −
1] =[ U − ∈ A δ (Σ B ⊗ K ) and that each U ( n ) is contained in S k ∈ N U + k (Σ B ). We have g Σ Bδ ([ U ( n )]) n ∈ N = [˜ g ( U ( n )) ] where ˜ g ( U ( n )) : Σ → A δ (Σ B ⊗ K ) is determined by theproperty ˜ g ( U ( n )) ( ω ) = [ n U ( n ) −
1] = [ U −
1] = [ G ] = h ( ω ) . Hence, g Σ Bδ ([ U ( n )]) n ∈ N = [˜ g ( U ( n )) ] = [ h ] and g Σ Bδ is surjective.Next suppose that ( u n ) n ∈ N is a sequence of unitaries in S k ∈ N U + k (Σ B ) such that g Σ Bδ ([ U ( n )]) n ∈ N = 0 ∈ [[Σ , Σ B ⊗ K ]] δ . Thus, there exists a discrete asymptotic homo-topy H : Σ → A δ I (Σ B ⊗ K ) with A ev ◦ H = ˜ g ( u n ) and A ev ◦ H = 0. As above,we may write H ( ω ) = [ U −
1] where U : N → ( I (Σ B ⊗ K )) + is a unitary-valued mapwith U ( n ) − ∈ I (Σ B ⊗ K ) for all n ∈ N . By assumption, lim n →∞ k U ( n )(0) − u n k = lim n →∞ k U ( n )(1) − k = 0. A standard argument shows that therefore[ u n ] = [ U ( n )(0)] = [ U ( n )(1)] = [1] = 0 ∈ K (Σ B ) if n is sufficiently large. Thus,([ u n ]) n ∈ N ∈ L n ∈ N K (Σ B ).On the other hand, if ([ u n ]) n ∈ N ∈ L n ∈ N K (Σ B ) then ˜ g ( u n ) ( ω ) = [ n u n −
1] = [ n
0] = 0, so that g Σ Bδ ([ u n ]) n ∈ N = 0. This completes the calculation of ker g Σ Bδ .Finally, it remains to prove that g Σ Bδ is additive. Thus, let ( u n ) n ∈ N and ( v n ) n ∈ N betwo sequences in S k ∈ N U + k (Σ B ). It is a well-known fact that [ u n ] + [ v n ] = [ u n v n ] =[ u n ∗ v n ] ∈ K (Σ B ), where u n ∗ v n is the concatenation of u n and v n , viewed as elementsof Σ( M k ( B ) + ). In particular, g Σ Bδ ([ u n ] + [ v n ]) n ∈ N = [˜ g ( u n ∗ v n ) ], and by definition of thegroup structure on [[Σ , Σ B ⊗ K ]] δ we obtain that indeed [˜ g ( u n ∗ v n ) ] = [˜ g ( u n ) ] + [˜ g ( v n ) ] = g Σ Bδ ([ u n ]) n ∈ N + g Σ Bδ ([ v n ]) n ∈ N .Note that Proposition 3.2, together with Bott periodicity for K-theory and for D-theoryand the stability isomorphism for D-theory, yields a chain of isomorphisms Q n ∈ N K ( B ) L n ∈ N K ( B ) ∼ = Q n ∈ N K (Σ B ) L n ∈ N K (Σ B ) ∼ = [[Σ , Σ B ⊗ K ]] δ ∼ = [[Σ ⊗ K , Σ B ⊗ K ⊗ K ]] δ = D (Σ , Σ B ⊗ K ) ∼ = D (Σ , B ) . However, we will give a more succinct description of this isomorphism next. Suppose forthe moment that B is a unital C*-algebra. Then we defineΨ B : Y n ∈ N K ( B ) → D (Σ , B )as follows: We write an element of Q n ∈ N K ( B ) as ([ p n ]) n ∈ N where each p n is a projec-tion in M ∞ ( B ) = S k ∈ N M k ( B ). We consider the discrete asymptotic homomorphism f ( p n ) : C → A δ ( B ⊗ K ) which is determined by f ( p n ) (1) = [ n p n ]. Now we define8 B by the prescription Ψ B ([ p n ]) n ∈ N = [Σ f ( p n ) ⊗ id K ] ∈ [[Σ ⊗ K , Σ B ⊗ K ⊗ K ]] δ = D (Σ , B ⊗ K ) ∼ = D (Σ , B ). Of course,Σ f ( p n ) ⊗ id K ( φ ⊗ ψ ⊗ T ) = [ n φ ⊗ ψ ⊗ p n ⊗ T ]for all φ, ψ ∈ Σ and T ∈ K . If B is non-unital we define Ψ B by requiring that thediagram 0 Q n ∈ N K ( B ) Q n ∈ N K ( B + ) Q n ∈ N K ( C ) 00 D (Σ , B ) D (Σ , B + ) D (Σ , C ) 0 Ψ B Ψ B + Ψ C with exact rows commutes. Theorem 3.3.
For any C*-algebra B , the map Ψ B : Q n ∈ N K ( B ) → D (Σ , B ) is anatural surjective group homomorphism with ker Ψ B = L n ∈ N K ( B ) .Proof. It is clear that Ψ B is natural in B . The proof consists of several parts: First wewill assume that B is unital in order to give another description of the map Ψ B . Bynaturality, we can extend this description to non-unital C*-algebras. Secondly we willuse this alternative description for the C*-algebra Σ B in order to prove the statementof the theorem for double suspensions Σ B . Finally we will use the concrete descriptionof Cuntz’s version of Bott periodicity [Cun84] to reduce the general case to the case ofdouble suspensions.We define a map ¯Ψ B : Q n ∈ N K ( B ) → D (Σ , B ⊗ K ) to be the composition Y n ∈ N K ( B ) β −→ Y n ∈ N K (Σ B ) g Σ Bδ −−→ [[Σ , Σ B ⊗ K ]] Σ −⊗ id K −−−−−→→ [[Σ ⊗ K , Σ B ⊗ K ⊗ K ]] δ = D (Σ , B ⊗ K ) ∼ = D (Σ , B ) , where β is the Bott periodicity isomorphism. It is clear that ¯Ψ B is natural in B . Wewill prove that Ψ B = ¯Ψ B . By naturality, it suffices to prove this for unital C*-algebras B .Thus, let B be a unital C*-algebra, and let p n ∈ S k ∈ N M k ( B ) be a sequence of projec-tions, representing an element ([ p n ]) n ∈ N ∈ Q n ∈ N K ( B ). By the description of the Bottperiodicity isomorphism K ( B ) → K (Σ B ), we have β ([ p n ]) n ∈ N = ([ ω ⊗ p n + 1]) n ∈ N . Inparticular, g Σ Bδ ◦ β ([ p n ]) n ∈ N = [˜ g ] where ˜ g : Σ → A δ (Σ B ⊗ K ) is such that ˜ g ( ω ) = [ n ω ⊗ p n ]. Of course, this implies that ˜ g ( ψ ) = [ n ψ ⊗ p n ] for all ψ ∈ Σ. In particular,Σ˜ g ⊗ id K : Σ ⊗K → A δ (Σ B ⊗K⊗K ) is such that Σ˜ g ⊗ id K ( φ ⊗ ψ ⊗ T ) = [ n φ ⊗ ψ ⊗ p n ⊗ T ]for all φ, ψ ∈ Σ and T ∈ K . Therefore, Σ˜ g ⊗ id K = Σ f ( p n ) ⊗ id K , which implies thatindeed Ψ B = ¯Ψ B . 9ow we consider a double suspension Σ B . In this case, ¯Ψ B can also be written as thecomposition Y n ∈ N K (Σ B ) β −→ Y n ∈ N K (Σ B ) g Σ3 Bδ −−−→ [[Σ , Σ B ⊗ K ]] δ ∼ = [[Σ , Σ B ⊗ K ]] −⊗ id K −−−−→ [[Σ ⊗ K , Σ B ⊗ K ⊗ K ]] = D ( C , Σ B ⊗ K ) Σ −→ D (Σ , Σ B ⊗ K ) ∼ = D (Σ , Σ B ) , where all maps in this composition are isorphisms, except for g Σ Bδ which is surjectivewith kernel equal to L n ∈ N K (Σ B ) by Proposition 3.2. Since β − ( L n ∈ N K (Σ B )) = L n ∈ N K (Σ B ), this implies the claim of the theorem for Σ B .In the case of general B , we consider the diagram0 L n ∈ N K (Σ B ) Q n ∈ N K (Σ B ) D (Σ , Σ B ) 00 L n ∈ N K ( B ) Q n ∈ N K ( B ) D (Σ , B ) 0 ∼ = Ψ S B ∼ = ∼ =Ψ B where the vertical arrows are the periodicity isomorphisms coming from Cuntz’s [Cun84]version of Bott periodicity. Recall that these periodicity isomorphisms are the indexmaps associated to a certain short exact sequence of C*-algebras. By construction ofthese index maps, the diagram above commutes because the horizontal maps are givenby natural transformations. We have already seen that the top row in the diagram isexact, so the bottom row must be exact as well. As mentioned in the introduction, the calculation D (Σ , A ) ∼ = Q n ∈ N K ( A ) L n ∈ N K ( A )implies that there are two different ways for defining the product of an element of Q n ∈ N K ( A ) / L n ∈ N K ( A ) with an E-theory class in E ( A, B ), yielding an element of Q n ∈ N K ( B ) / L n ∈ N K ( B ). The following result states that these two products are infact equal. Theorem 4.1.
Let A and B be C*-algebras and fix η ∈ E ( A, B ) . Then the compositions Q n ∈ N K ( A ) L n ∈ N K ( A ) ∼ = Q n ∈ N E ( C , A ) L n ∈ N E ( C , A ) → Q n ∈ N E ( C , B ) L n ∈ N E ( C , B ) ∼ = Q n ∈ N K ( B ) L n ∈ N K ( B )10 nd Q n ∈ N K ( A ) L n ∈ N K ( A ) ∼ = D ( S C , A ) → D ( S C , B ) ∼ = Q n ∈ N K ( B ) L n ∈ N K ( B ) , which are given by the respective composition product with η , coincide. Before we step into the proof of the theorem, we state a lemma that we will need in thecourse of the proof.
Lemma 4.2.
Let B be a C*-algebra, and fix τ ∈ [0 , . Assume that ˜ E ⊂ A IB isseparable, and put ˜ E τ = A ev τ ( ˜ E ) ⊂ A B . Let E ⊂ T IB be separable with as IB ( E ) =˜ E , and put E τ = T ev τ ( E ) ⊂ T B (so that in particular as B ( E τ ) = ˜ E τ ). Let r : P → P be a reparametrization which is admissible for both E and E τ . Then the diagram ˜ E A IB ˜ E τ A B Φ A ev τ A ev τ Φ commutes if we use r to define both horizontal maps Φ .Proof. Let F ∈ E be arbitrary. Then Φ[ π ◦ F ] = [ t F ( t )( r ( t ))] ∈ A IB and hence A ev τ ◦ Φ[ π ◦ F ] = [ t F ( t )( r ( t ))( τ )] ∈ A B. On the other hand, A ev τ [ π ◦ F ] = [ π ◦ F τ ] where F τ = T ev τ ( F ) ∈ E τ . Therefore,Φ ◦ A ev τ [ π ◦ F ] = Φ[ π ◦ F τ ] = [ t F τ ( t )( r ( t ))]= [ t F ( t )( r ( t ))( τ )] = A ev τ ◦ Φ[ π ◦ F ]as claimed. Proof of Theorem 4.1.
By a naturality argument, we may assume without loss of gen-erality that A and B are unital. Let ( p n ) n ∈ N be a sequence of projections in A ⊗ K ,and represent η by an asympttic homomorphism f : A → A B . Then the image of[([ p n ]) n ∈ N ] ∈ Q n ∈ N K ( A ) / L n ∈ N K ( A ) under the first composition is represented by afamily (˜ p n ) n ∈ N of projections in B ⊗K which have the property that the asymptotic homo-topy classes of the asymptotic homomorphisms φ [ t φ ⊗ ˜ p n ] and φ f ( φ ⊗ p n ) agree.Let H n : Σ → A I (Σ B ⊗ K ) be asymptotic homotopies with A ev ◦ H n ( φ ) = f ( φ ⊗ p n )and A ev ◦ H n ( φ ) = [ t φ ⊗ ˜ p n ] for all φ ∈ Σ.We have to prove that the second composition in the statement of the theorem maps[([ p n ]) n ∈ N ] to [([˜ p n ]) n ∈ N ] as well. It follows from the description of Ψ A in Theorem 3.3that under the identification Q n ∈ N K ( A ) L n ∈ N K ( A ) ∼ = [[Σ , Σ A ⊗ K ]] δ ∼ = [[Σ , Σ A ⊗ K ]] , p n ]) n ∈ N ] is first mapped to the class of the discrete asymptotic homomorphism ψ ⊗ φ [ n ψ ⊗ φ ⊗ p n ], and then to the class of the sequentially trivial asymptotichomomorphism g : Σ → A (Σ A ⊗ K ) which is given by g ( ψ ⊗ φ ) = [ t ψ ( t − ⌊ t ⌋ ) φ ⊗ p ⌊ t ⌋ ] . Analogously, [([˜ p n ]) n ∈ N ] is identified with the class of the sequentially trivial homomor-phism ˜ g : Σ → A (Σ B ⊗ K ) which is given by ˜ g ( ψ ⊗ φ ) = [ t ψ ( t − ⌊ t ⌋ ) φ ⊗ ˜ p ⌊ t ⌋ ]. Thus,we have to prove that the sequentially trivial asymptotic homomorphisms f • g and ˜ g are asymptotically homotopic. For appropriate choices in the respective definitions of Φwe have f • g ( ψ ⊗ φ ) = Φ (cid:0) A f [ t ψ ( t − ⌊ t ⌋ ) φ ⊗ p ⌊ t ⌋ ] (cid:1) = Φ (cid:2) t ψ ( t − ⌊ t ⌋ ) f ( φ ⊗ p ⌊ t ⌋ ) (cid:3) = Φ (cid:2) t ψ ( t − ⌊ t ⌋ ) A ev ◦ H ⌊ t ⌋ ( φ ) (cid:3) = Φ ◦ A ev (cid:2) t ψ ( t − ⌊ t ⌋ ) H ⌊ t ⌋ ( φ ) (cid:3) = A ev ◦ Φ (cid:2) t ψ ( t − ⌊ t ⌋ ) H ⌊ t ⌋ ( φ ) (cid:3) , where the last equality is due to Lemma 4.2. Of course, this is asymptotically homotopicto the discrete asymptotic homomorphism ψ ⊗ φ
7→ A ev ◦ Φ[ t ψ ( t − ⌊ t ⌋ ) H ⌊ t ⌋ ( φ )], andagain by Lemma 4.2 we have A ev ◦ Φ (cid:2) t ψ ( t − ⌊ t ⌋ ) H ⌊ t ⌋ ( φ ) (cid:3) = Φ (cid:2) t ψ ( t − ⌊ t ⌋ ) A ev ◦ H ⌊ t ⌋ ( φ ) (cid:3) = Φ (cid:2) t ψ ( t − ⌊ t ⌋ ) (cid:2) s φ ⊗ ˜ p ⌊ t ⌋ (cid:3)(cid:3) = (cid:2) t ψ ( t − ⌊ t ⌋ ) φ ⊗ ˜ p ⌊ t ⌋ (cid:3) = ˜ g ( ψ ⊗ φ ) . Thus, indeed f • g and ˜ g are asymptotically homotopic.There is a notable special case of Theorem 4.1. In order to state it, we recall that for any j ∈ N , the j -th K-homology group of a C*-algebra B is given by K j ( B ) = E ( B, Σ j C ).In particular, if η ∈ K j ( B ) is a K-homology class and ξ ∈ K ℓ ( B ⊗ A ) ∼ = E ( C , Σ ℓ ( B ⊗ A ))is a K-theory class, we can define the Kronecker pairing of η and ξ to be h η, ξ i = (Σ ℓ η ⊗ id A ) • ξ ∈ E ( C , Σ ℓ + j A ) ∼ = K ℓ + j ( A ) . Now suppose that ( ξ n ) n ∈ N is a sequence in K ℓ ( B ⊗ A ). This sequence then definesa class in Q n ∈ N K ℓ ( B ⊗ A ) / L n ∈ N K ℓ ( B ⊗ A ) ∼ = D (Σ , Σ ℓ ( B ⊗ A )), and the sequenceof Kronecker pairings ( h η, ξ n i ) n ∈ N likewise defines an element of D (Σ , Σ ℓ + j A ). NowTheorem 4.1 directly implies the following: Corollary 4.3.
In this situation, Ψ Σ ℓ + j A [( h η, ξ n i ) n ∈ N ] = (Σ ℓ η ⊗ id A ) • Ψ Σ ℓ ( B ⊗ A ) [( ξ n ) n ∈ N ] ∈ D (Σ , Σ ℓ + j A ) , where the right-hand side is defined using the composition product E (Σ ℓ ( B ⊗ A ) , Σ ℓ + j A ) × D (Σ , Σ ℓ ( B ⊗ A )) → D (Σ , Σ ℓ + j A ) . Thus, it is possible to use the D-theory product to calculate Kronecker pairings asymp-totically. 12 eferences [BK97] Bruce Blackadar and Eberhard Kirchberg. “Generalized inductive limits offinite-dimensional C ∗ -algebras”. In: Math. Ann.
J. Noncommut. Geom. K -th´eorie bivariante”. In: C. R. Acad. Sci. Paris S´er. I Math. K -theory and C ∗ -algebras”. In: Algebraic K -theory, numbertheory, geometry and analysis (Bielefeld, 1982) . Vol. 1046. Lecture Notes inMath. Springer, Berlin, 1984, pp. 55–79.[GHT00] Erik Guentner, Nigel Higson, and Jody Trout. “Equivariant E -theory for C ∗ -algebras”. In: Mem. Amer. Math. Soc. K -theory and the Novikov conjecture”. In: Geom.Funct. Anal. K -theory for groups whichact properly and isometrically on Hilbert space”. In: Electron. Res. Announc.Amer. Math. Soc.
J. Differential Geom. K -Theory Geom. Dedicata