Homology, Homotopy and Applications, vol.2, No.10, 2000, pp.127–145 ISSN 1512–0139

KK-THEORY AS THE K-THEORY OF C^{∗}-CATEGORIES

TAMAZ KANDELAKI

(communicated by Jonathan Rosenberg)

Abstract Let complex C^{∗} algebras be endowed with a norm-continuous action of a fixed compact second countable group. From a separable C^{∗}- algebra A and a σ-unital C^{∗}-algebra B, we construct a C^{∗}-category Rep(A, B) and an isomorphism

κ : K^{i+1}(Rep(A, B)) → KK^{i}(A, B), i ∈ ℤ_{2},

where on the left-hand side are Karoubi’s topological K-groups, and on the right-hand side are Kasparov’s equivariant bivariant K-groups.

1. Introduction

The purpose of this article is to study the possibility of calculation of Kasparov KK-theory by K-theory. Some partial results are known in this direction, for example: Paschke’s result on K-homology of nuclear C^{∗}-algebras [13], the generalization of Paschke’s theorem for Kasparov KK-groups when the first argument is nuclear [7], [17], Higson’s modification of Paschke’s result for K-homology of separable C^{∗}-algebras [6], and Künneth type theorem results for KK-theory [15]. In all these situations, the algebras are trivially graded.

Let us present briefly the idea of this paper. The main objects of our study are additive C^{∗}-categories Rep(A, B) and Rep(A, B), where A and B are trivially graded C^{∗}-algebras with fixed compact group actions. In the first category, objects are equivariant A, B-bimodules and morphisms are invariant B-homomorphisms which commute, up to the ideal of compact homomorphisms, with the action of A. After definition of the first category, we define the category Rep(A, B) as the universal pseudoabelian C^{∗}-category of Rep(A, B). (The notation ‘universal pseudoabelian’ is slightly different from Karoubi’s analogous definition [8], [9]). After small modification of Karoubi’s K-theory of a Banach category for C^{∗}-categories, we study properties of the K-groups of Rep(A, B). Then we apply this to prove our main result, that the K-groups of Rep(A, B) are essentially isomorphic to Kasparov’s equivariant KK-groups, up to a dimension shift, when A is a separable C^{∗}-algebra and B is a σ-unital C^{∗}-algebra with fixed compact group actions.

This article is organized as follows. In Section 1 we review the basic definitions and properties of C^{∗}-categories [4]. We give a construction of the universal pseudoabelian C^{∗}-category of an additive C^{∗}-category, and a characterization of a cofinal subcategory of H(B ⊗C^{(1,0)}), that are used in the next sections. In Section 2 we review Karoubi’s results ([8], [9]) on K-theory of Banach categories, adapted specially for C^{∗}-categories. In Section 3 we give some remarks on the definition of the KK-groups in the form that is used in the sequel, and especially a characterization of the KK-groups in the case when the algebras are trivially graded [2], [10], [16]. In Section 4 we prove our main theorem.

Note that our main result shows that the category Rep(A, B) is an interesting object to be studied from various points of view (particularly, for the study of algebraic K-theory, cyclic

The work was partially supported by Grant No GM1-115 of the U.S. CRDF, INTAS Georgia Grant No 213. Received 19 February 1999, revised 7 November 2000; published on 14 November 2000. 2000 Mathematics Subject Classification: 19K35, 19K99, 46L89, 46M99. Key words and phrases: K-theory, KK-theory, C^{∗}-category. \begin{matrix}c \\ ◯ \end{matrix} 2000, Tamaz Kandelaki. Permission to copy for private use granted.

homology and connections with K-theory, i.e., connections with KK-theory).

I would like to express thanks to Prof. H. Inassaridze for his care during the working on this article, Prof. J. Rosenberg for his interest and attention. Especially I am thankful to the referee for his valuable comments and his hard work on the improving my language and which made the article readable.

2. Remarks on C^{∗}-categories

In this section we recall the definition of a C^{∗}-category, and the main properties and examples, which are used in the next sections.

2.1. Definition and some properties of a C^{∗}-category By ℂ is denoted the field of complex numbers.

Definition 1. A ℂ-linear category A is called a C^{∗}-category if :

a) hom(a, b) is a complex Banach space, the composition of morphisms is bilinear and ∥fg∥ ⩽ ∥f∥ · ∥g∥ for every pair of composable morphisms f and g;

b) There is an involutive antilinear contravariant endofunctor ∗ : A → A, which preserves objects.

c) ∥f∥^{2} = ∥f^{∗}f∥ for each morphism f, where f^{∗} = ∗(f);

d) The morphism f^{∗}f is a positive element of the C^{∗}-algebra hom(a, a) for each f ∈ hom(a, b).

Example 2. 1) The category with Hilbert spaces as objects and all bounded linear maps as morphisms is a C^{∗}-category, which will be denoted by H.

2) Let B be a C^{∗}-algebra. The category with right Hilbert B-modules as objects and all bounded B-homomorphisms, which have an adjoint, as morphisms is a C^{∗}-category. We denote it by H(B). If E and F are modules from H(B) then L(E, F) or hom(E, F) denotes the space of morphisms from E to F, and K(E, F) denotes the ideal of compact B-homomorphisms from E to F.

3) A unital C^{∗}-algebra is a C^{∗}-category with one object and the elements of the algebra themselves as morphisms.

Definition 3. Let A and B be C^{∗}-categories. A functor F : A → B is said to be a ∗-functor if

a) F(f + g) = F(f) + F(g);

b) F(λf) = λF(f);

c) F(f^{∗}) = F(f)^{∗}.

∗-functors between C^{∗}-categories, like ∗-homomorphisms of C^{∗}-algebras, are norm-decreasing. For the following theorem we refer to [4].

Theorem 4. Every C^{∗}-category A may be realized as a concrete C^{∗}-category, i.e., there is a faithful ∗-embedding F : A → H

Let A be a C^{∗}-category and I ⊂ Hom A a subset. Put hom(a, b)_{I} = hom(a, b) ∩ I. Then I is called a left ideal if hom(a, b)_{I} is linear subspace of hom(a, b) and f ∈ hom(a, b)_{I}, g ∈ hom(b, c) imply gf ∈ hom(a, c)_{I}. A right ideal is defined similarly. I is two-sided ideal if it is both left and right ideal. An ideal I is closed if hom(a, b)_{I} is closed in hom(a, b) for each pair of objects.

I determines an equivalence relation on the morphisms of A: f ∼ g if f − g ∈ I. If I = I^{∗} is an ideal of A, the set of equivalence classes A/I can be made into a C^{∗}-category in a unique way by requiring that the canonical map f ↦ f̂ give rise to a ∗-functor A → A/I. A/I can be made into a normed ∗-category, by defining ∥f̂∥ = sup_{g∈f̂}∥g∥. Arguing as for C^{∗}-algebras, one can show Proposition 5. Let A be a C^{∗}-category and I a be closed, two-sided ideal of A. Then I = I^{∗} and A/I is a C^{∗}-category.

Example 6. ¿From Example 2(2) it follows that there exists a C^{∗}-category Cal(B) = H(B)/K(B) which sometimes will be called the Calkin C^{∗}-category over B.

A word about ℤ_{2}-graded C^{∗}-categories: Let A be a C^{∗}-category. A ℤ_{2}-grading on A is a direct sum decomposition, for any pair of objects a, b ∈ A, hom(a, b) = hom^{(0)}(a, b) ⊕ hom^{(1)}(a, b), with hom^{(0)}(a, b) and hom^{(1)}(a, b) two closed linear subspaces of hom(a, b), such that

a) if f ∈ hom^{(i)}(a, b) and g ∈ hom^{(j)}(b, c), then gf ∈ hom^{(i+j)}(a, c);

b) if f ∈ hom^{(i)}(a, b), then f^{∗} ∈ hom^{(i)}(b, a).

A morphism from hom^{(i)}(a, b) is called homogeneous of degree i. The degree of a homogeneous element f is denoted ∂f.

A ∗-functor F : A → B of graded C^{∗}-categories A and B is graded if F(hom^{(i)}(a, b)) ⊂ hom^{(i)}(F(a), F(b)) for any pair of objects a, b from A.

Let Γ : A → A be a ∗-functor, which is the identity map on objects and such that Γ^{2} = id_{A}. Then hom^{(0)}(a, b) = {f : F(f) = f} and hom^{(1)}(a, b) = {f : F(f) = −f} gives a ℤ_{2}-grading on A. Conversely, given a grading, one can define a corresponding ∗-functor Γ by the identity Γ(f^{(0)} + f^{(1)}) = f^{(0)} − f^{(1)}.

Let F : A → B and G : A → B be graded ∗-functors. A set

α = {α_{a} : F(a) → G(a)}_{a∈Ob A}

of morphisms is called a natural transformation of degree i, if ∂α_{a} = i for all α_{a} and

G(f)α_{a} = (−1)^{∂α∂f}α_{b}F(f)

for any homogeneous morphism f : a → b from A.

A natural transformation α : G → F is called bounded if sup_{a} ∥α_{a}∥ < ∞. Hereafter, by ‘transformation’ we will always mean ‘bounded transformation.’

Example 7. Let B be a ℤ_{2}-graded σ-unital algebra and let ℍ(B) be the ℤ_{2}-graded C^{∗}- category with countably generated ℤ_{2}-graded right Hilbert B-modules as objects, and Bhomomorphisms between Hilbert modules of degree i ∈ ℤ_{2}, that have an adjoint, as the morphisms of degree i. Let E = E^{(0)}⊕E^{(1)} be a module from ℍ(B). Denote by Ě = Ě^{(0)}⊕Ě^{(1)} the opposite graded module to E, Ě^{(0)} = E^{(1)} and Ě^{(1)} = E^{(0)}. Next we need the following endofunctor and natural transformation of degree 1.

Let 𝕍 : ℍ(B) → ℍ(B) be the covariant functor defined by the formula 𝕍(E) = Ě and 𝕍(f) = (−1)^{∂f}f, and consider natural transformation τ : id_{ℍ(B)} → 𝕍 of degree 1 given by morphisms τ_{E} : E → Ě:

τ_{E} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ,

in the decomposition Ě = Ě^{(0)} ⊕ Ě^{(1)}. One checks that τ_{E}^{∗} = −τĚ.

2.2. Additive and pseudoabelian C^{∗}-categories

First we recall that a projection in a C^{∗}-category is a morphism with the properties p^{∗} = p and p^{2} = 1, i.e., a projection is a self-adjoint idempotent.

Definition 8. An additive C^{∗}-category is said to be a pseudoabelian C^{∗}-category if each projection has a kernel.

Remark. The main difference from the analogous definition of [8] and [9] is that here idempotents in addition are self-adjoint, i.e., are projections.

The following theorem describes how an additive C^{∗}-category can be embedded in a pseudoabelian C^{∗}-category (cf. [8], [9]).

Theorem 9. Let A be an additive C^{∗}-category. There exists a pseudoabelian C^{∗}-category Ã and an additive ∗-functor ϕ : A → Ã with the following universal property. For any pseudoabelian C^{∗}-category D and any additive ∗-functor ψ : A → D there exists a unique additive ∗-functor ψ′ : Ã → D such that ψ = ϕ · ψ′. The pair (ϕ, Ã) is unique up an additive ∗- equivalence of additive C^{∗}-categories.

Proof. We only give here the constructions, because the proofs are precisely analogous to those in [8], [9]. An object of Ã has the form (E, p), where E ∈ Ob(A) and p ∈ hom(E, E) is a projection. A morphism from (E, p) to (F, q) is defined as a morphism f : E → F of A such that fp = qf = f. The composition of morphisms is defined as a composition of morphisms in A. The sum of objects is given by formula (E, p) ⊕ (F, q) = (E ⊕ F, p ⊕ q), and the norm of morphisms is inherited from A. □

As suggested in [8] and [9], the construction of K-theory is based on the notion of pseudoabelianness of an additive category, and is slightly different from the similar definition given here. We carry out the construction using notion of a pseudoabelian C^{∗}-category. Consider A as a Banach category and denote by ξA the pseudoabelian category A in Karoubi’s sense. We have following:

Lemma 10. Let A be an additive C^{∗}-category. Then the category Ã is additively equivalent to ξA.

Proof. Let i : Ã → ξA be the faithful functor that is the identity on objects and morphisms. To define j : ξA → Ã firstly note that if q ∈ hom(F, F) is an idempotent then

q̄ = \sqrt{(2q^{∗} − 1)(2q − 1) + 1} · q · \sqrt{((2q^{∗} − 1)(2q − 1) + 1)^{−1}}

is a projection [10] and the pairs (F, q) and (F, q̄) are isomorphic by

u_{q} = \sqrt{(2q^{∗} − 1)(2q − 1) + 1}.

Then define j by the formulas (E, q) ↦ (E, u_{q}qu_{q}^{−1}) on objects and j(f) = u_{q′}fu_{q} for a morphisms, where f : (E, q) → (E′, q′). Now it is easy to show that the isomorphism i·j ≃ id_{ξA} is given by the essential isomorphisms {u_{q}} and j · i = idÃ. □

2.3. On the main examples of C^{∗}-categories

Here we assume that all C^{∗}-algebras are trivially graded and they have a norm-continuous action of a fixed compact group.

Having treated pseudoabelian C^{∗}-categories, we now proceed to one of the main examples of this paper.

Example 11. 1) Firstly we define the C^{∗}-category H(B) over fixed compact second countable group G. The objects of this category are all countable generated right Hilbert Bmodules equipped with a B-linear, norm-continuous G-action such that g(xb) = g(x)g(b) and < g(x), g(y) >= g < x, y >, for all g ∈ G. A morphism f : E → E′ is B-homomorphism such that there exists f^{∗} : E′ → E satisfying the conditions: < T (x), y >=< x, T^{∗}(y) > where x ∈ E and y ∈ E′. The norm of a morphism is defined as the norm of linear bounded map. It is easy to check that H(B) is an additive C^{∗}-category with respect to the sum of the Hilbert modules . Finally, note that compact group acts on the morphisms by the following way: if f : E → E′ then morphism gf : E → E′ is defined by the formula gf(x) = g(f(g^{−1}(x))). (Note that this action generally is not norm-continuous). A morphism is called invariant if gf = f. In the next, under H(B) we mean above constructed additive C^{∗}-category with this action of G (cf. [11])

2) Now, we define the C^{∗}-category rep(A, B). The objects of this category are pairs of form (E, ϕ), where E is a countably generated right Hilbert B-module and ϕ : A → L(E) is an equivariant ∗-homomorphism. Objects of this type are said to be A, B-bimodules (cf. [16]).

A morphism f : (E, ϕ) → (E′, ϕ′) is an invariant B-homomorphism f : E → E′ in H(B) such that fϕ(a) = ϕ′(a)f for all a ∈ A. The structure of C^{∗}-category is inherited from the C^{∗}-category structure of H(B) and it is easy to show that rep(A, B) is an additive C^{∗}-category (in fact, a pseudoabelian C^{∗}-category). The following property of rep(A, B) will be used to calculate the K-groups of rep(A, B). Note that there exists a ∗-functor ∞ : H(B) → H(B) and a natural isomorphism of functors id_{H(B)} ⊕ ∞ ≃ ∞, where E^{∞} = E ⊕ E ⊕ · · ·. This structure induces a corresponding structure on rep(A, B) via the formulas (E, ϕ)^{∞} = (E^{∞}, ϕ^{∞}), where ϕ^{∞}(a) = (ϕ(a))^{∞} for all a ∈ A. This structure will be called an ∞-structure.

3) Consider the additive C^{∗}-category Cal(B) which is the quotient C^{∗}-category H(B)/K(B). It has an essential compact group action inherited from the action of a compact group on H(B). Denote by π : H(B) → Cal(B) the canonical additive ∗-functor. We need also following C^{∗}- category denoted by Cal(A, B). By definition objects of this category have the form (E, ψ), where E is a Hilbert B-module and ψ : A → hom_{CalB}(E, E) is a equivariantly liftable ∗- homomorphism, i.e., there exists an A, B-bimodule (E, ϕ) such that πϕ = ψ. A morphism f : (E, ψ) → (E′, ψ′) is a morphism f : E → E′ of the category Cal(B) such that fψ(a) = ψ′(a)f for all a ∈ A, and has a invariant lifting in H(B). Define the ∗-functor Θ : rep(A, B) → Cal(A, B) by (E, ϕ) ↦ (E, πϕ) and f ↦ π(f).

4) Now, we want to define the additive C^{∗}-category Rep(A, B). The objects of this category are also A, B-bimodules, i.e., objects are pairs (E, ϕ), where E is a countably generated right Hilbert B-module and ϕ : A → L(E) is a equivariant ∗-homomorphism. Also, a morphism f : (E, ϕ) → (E′, ϕ′) is a invariant morphism f : E → E′ in H(B) such that

fϕ(a) − ϕ′(a)f ∈ K(E, E′)

for all a ∈ A. The structure of C^{∗}-category is inherited from H(B). It is easy to show that Rep(A, B) is an additive C^{∗}-category but it isn’t a pseudoabelian C^{∗}-category. There is a canonical additive ∗-functor Π_{A,B} : Rep(A, B) → Cal(A, B) defined by (E, ϕ) ↦ (E, πϕ) and f ↦ πf. From the definition follows easily that the canonical linear map

hom((E, ϕ), (E′, ϕ′)) ↦ hom((E, πϕ), (E, πϕ′)

is surjective, i.e., Π is a Serre functor (see for the definition [8]). □

Now we come to our main C^{∗}-category, that is, Rep(A, B).

Definition 12. The C^{∗}-category Rep(A, B) is the universal pseudoabelian C^{∗}-category of Rep(A, B). Using the definition of a pseudoabelian C^{∗}-category, we have the following description of Rep(A, B). Objects of it are triples (E, ϕ, p), where (E, ϕ) is an object and p : (E, ϕ) → (E, ϕ) is a morphism of Rep(A, B) such that p^{∗} = p and p^{2} = p. A morphism f : (E, ϕ, p) → (E′, ϕ′, p′) is a morphism f : (E, ϕ) → (E′, ϕ′) of Rep(A, B) such that fp = p′f = f. In detail, f has the properties

fϕ(a) − ϕ′(a)f ∈ K(E, F), fp = p′f = f.  (1)

The structure of C^{∗}-category of Rep(A, B) comes from the corresponding structure of Rep(A, B). In particular, the sum of triples is given by formula (E, ϕ, p) ⊕ (F, ψ, q) = (E ⊕ F, ϕ ⊕ ψ, p ⊕ q).

2.4. On a cofinal subcategory of ℍ(B ⊗ C^{(1,0)})

When A and B are trivially graded C^{∗}-algebras, for an interpretation of KK^{1}(A, B), we need information on the following subcategory of ℍ(B ⊗ C^{(1,0)}). We begin this subsection recalling a definition from [8], [9].

Definition 13. An additive ∗-functor F : A → B of additive C^{∗}-categories is said to be quasisurjective if for any object b from B there are objects a and b′ from A and B respectively, and a unitary isomorphism b ⊕ b′ ≃ F(a). In particular, an additive C^{∗} sub-category A of B is cofinal iff the canonical inclusion is quasi-surjective.

Let C^{(1,0)} be the Clifford algebra with one generator g (g^{∗} = g and g^{2} = 1), with trivial action of a compact group. Consider the cofinal full subcategory ℍ_{B}(B⊗C^{(1,0)}) of ℍ(B⊗C^{(1,0)}) which contains modules isomorphic to modules of form

E_{n+1,n} = E ⊗_{B⊗C^{(1,0)}} (C^{n,n} ⊗_{ℂ⊗C^{(1,0)}} C^{(1,0)})

where E is a trivially graded equivariant B-module.

Also,denote by ℍ_{C^{(1,0)}}(B) the full subcategory of ℍ(B) with objects isomorphic to Bmodules of the form

E′_{n+1,n} = E ⊗_{B⊗ℂ} (C^{n,n} ⊗ C^{(1,0)}).

There is a canonical additive ∗-functor

S : ℍ_{B}(B ⊗ C^{(1,0)}) → ℍ_{C^{(1,0)}}(B)

defined by formulas S(E_{n+1,n}) = E′_{n+1,n} and S(f) = f for every B ⊗ C^{(1,0)}-homomorphism f, because f may be considered also as a B-homomorphism. This functor is injective, i.e., the linear maps on hom sets are injective. Note that a B-homomorphism f : E′_{n+1,n} → F′_{n+1,n} defines a B⊗C^{(1,0)}-homomorphism iff the morphism f is invariant under the action of 1⊗1⊗ε, i.e.,

(1 ⊗ 1 ⊗ ε)f(1 ⊗ 1 ⊗ ε) = f.

But the C^{∗}-category ℍ_{C^{(1,0)}}(B) coincides with the full subcategory of ℍ(B) generated by modules isomorphic to E ⊕ Ě, where E is trivially graded equivariant B-module. For each E ⊕ Ě consider the element

ε_{E} = ( \begin{matrix}0 & −τĚ \\ τ_{E} & 0 \end{matrix} )  (2)

where τ_{E} : E → Ě is the canonical isomorphism of degree 1. (See example 7.) Then ε_{E}^{∗} = ε_{E} and ε_{E}^{2} = 1. Consider the essential ℤ_{2}-action on the ℍ_{C^{(1,0)}}(B) defined as follows. Let f : E ⊕ Ě → F ⊕ F̌ be a B-homomorphism. Then we have

( \begin{matrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{matrix} ) ↦ ( \begin{matrix}f̌_{22} & f̌_{21} \\ f̌_{12} & f̌_{11} \end{matrix} )  (3)

where f̌ = 𝕍(f). (See example 7.) In particular, if f is a homomorphism of degree 0 invariant under this action, then

f = ( \begin{matrix}f_{0} & 0 \\ 0 & f̌_{0} \end{matrix} )  (4)

and if f has degree 1 under this action then

f = ( \begin{matrix}0 & f̌_{1} \\ f_{1} & 0 \end{matrix} ) .  (5)

Denote by ℍ_{∗}(B) this invariant subcategory of ℍ_{C^{(1,0)}}(B). Thus we have the following:

Proposition 14. Let B be a trivially graded C^{∗}-algebra. Then the graded additive C^{∗}-category ℍ_{B}(B ⊗ C^{(1,0)}) is graded additively isomorphic to ℍ_{∗}(B).

3. K-groups of an additive category

The purpose of this section is to transform some main results of K-theory of Banach categories, introduced by M. Karoubi in [8], [9], to a C^{∗}-category. There are some minor changes which will be needed in sequel sections.

3.1. K^{0} and K^{1} groups for an additive C^{∗}-category Definition 15. The group K^{0}(A) of an additive C^{∗}-category is the Grothendieck group of the abelian monoid of unitary isomorphism classes of objects of A.

Note that this definition coincides with usual definition because in a C^{∗}-category, objects are isomorphic if and only if they are unitarily isomorphic. Indeed, if u : E → F is an isomorphism, then u_{0} = u\sqrt{(u^{∗}u)^{−1}} is a unitary isomorphism. So from lemma 10 we get the following:

Proposition 16. Let A be an additive C^{∗}-category. The canonical functor induces an isomorphism i_{∗} : K^{0}(Ã) → 𝕂^{0}(ξA), where the left-hand K-group is as in the definition above, and the right one as in [8], [9].

Now we discuss analogous questions for the K^{−1} group (cf. [8], [9]).

Definition 17. Let A be an additive C^{∗}-category. Consider the set of pairs (E, α), where E ∈ ObA and α ∈ hom(E, E) is a unitary automorphism.

a). The pairs (E, α) and (E′, α′) are said to be unitarily isomorphic if there exists a unitary isomorphism u : E → E′ such that diagram

\begin{matrix}E & →^{u} & E′ \\ ↓^{α} & & ↓^{α′} \\ E & →^{u} & E′ \end{matrix}

is commutative.

b). The pairs (E, α) and (E, α′) are said to be homotopic if α and α′ are homotopic in Aut E.

c). A pair (E, α) is said to be elementary if it is homotopic to (E, 1_{E}).

d). The sum is defined by the formula

(E, α) ⊕ (E′, α′) = (E ⊕ E′, α ⊕ α′).

e). The pairs (E, α) and (E, α′) are said to be stably isomorphic if there exist elementary pairs (Ē, ē) and (Ê, ê), and a unitary isomorphism

(E, α) ⊕ (Ē, ē) ≃ (E′, α′) ⊕ (Ê, ê).

f). The abelian monoid K^{−1}(A) is defined as the monoid of classes of stably isomorphic pairs. Denote by d(E, α) the class of (E, α) in K^{−1}(A).

Lemma 18. (Cf. [9].) There are the following relations in K^{−1}(A):

a). d(E, α) + d(E, α^{∗}) = 0;

b). If α and α′ are homotopic unitary isomorphisms, then d(E, α) = d(E, α′).

c). d(E, α) + d(E, β) = d(E, βα);

In particular, K^{−1}(A) is an abelian group.

Proof. a). The unitary automorphism α ⊕ α^{∗} can be written in the form

( \begin{matrix}0 & α \\ α^{∗} & 0 \end{matrix} ) · ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

but each matrix is homotopic to 1_{E⊕E}. Thus a) holds.

b). Apply a). We get

d(E, α′) − d(E, α) = d(E ⊕ E, α′ ⊕ α^{∗})

But α′ ⊕ α^{∗} is homotopic to α ⊕ α^{∗}. Thus d(E ⊕ E, α′ ⊕ α^{∗}) = 0.

c). Note that d(E, α) + d(E, β) = d(E ⊕ E, α ⊕ β) and

d(E, αβ) = d(E ⊕ E, αβ ⊕ 1_{E}).

But (α ⊕ β)^{∗}(αβ ⊕ 1_{E}) = β ⊕ β^{∗}, which is homotopic to 1_{E⊕E}. Thus α ⊕ β is homotopic to αβ. Thus we can apply b). □

The next proposition is analogous to the corresponding property of the group K^{0}(A).

Proposition 19. Let A be an additive C^{∗}-category. The canonical homomorphism

i_{∗} : K^{−1}(A) → 𝕂^{−1}(A),

defined by d(E, α) ↦ d(E, α) is an isomorphism. Here 𝕂^{−1}(A) is Karoubi’s group.

Proof. i_{∗} is an epimorphism: Let (E, α) be a pair with α an isomorphism. Consider the unitary isomorphism ᾱ = α\sqrt{α^{∗}α}. It is homotopic to α, because α^{∗}α is homotopic to 1_{E}. Apply the lemma. We get that d(E, α) = d(E, ᾱ). i is a monomorphism: If i(d(E, α)) = 0, then there exists elementary (E′, e′) such that (E ⊕ E′, α ⊕ e′) is elementary. Then (E ⊕ E′, \overline{α ⊕ e′}) is also elementary, that is (E ⊕ E′, α ⊕ ē′) elementary. This means d(E, α) = 0. □

Thus the properties of K^{−1}(A) are inherited from the corresponding properties of 𝕂^{−1}(A). In particular, we get the following:

Lemma 20. d(E, α) = 0 if there exists an object G such that α ⊕ 1_{G} is homotopic to 1_{E⊕G}.

Theorem 21. Let A be an additive C^{∗}-category, Ã be the associated pseudoabelian C^{∗}-category and i : A → Ã the canonical additive ∗-functor. Then the induced homomorphism

i_{∗} : K^{−1}(A) → K^{−1}(Ã)  (6)

is an isomorphism.

3.2. The K-group of a ∗-functor Definition 22. Let A and B be additive C^{∗}-categories and F : A → B be an additive ∗- functor. Denote by Γ(F) the set of triples (E, F, α), where E and F are objects of A, and α : F(E) → F(F) is a unitary isomorphism from B.

a) Two triples (E, F, α) and (E′, F′, α′) are unitarily isomorphic if there exist unitary isomorphisms f : E → E′ and g : F → F′ such that the diagram

F(E) →^{α} F(F)
↓^{F(f)}  ↓^{F(g)}
F(E′) →^{α′} F(F′)

is commutative.

b). Two triples (E, F, α) and (E, F, α′) are called homotopic if α and α′ are homotopic in the subspace of unitary isomorphisms of hom(E, F).

c). The triple (E, E, 1_{E}) is called trivial. A triple (E, F, α) is said to be elementary if this triple is homotopic to the trivial triple.

e). The sum of triples is defined by the formula (E, F, α) ⊕ (E′, F′, α′) = (E ⊕ E′, F ⊕ F′, α ⊕ α′).

f). Two triples σ = (E, F, α) and σ′ = (E′, F′, α′) are stably unitarily isomorphic if there exist elementary pairs τ = (Ē, Ē, ᾱ) and τ′ = (Ē′, Ē′, ᾱ′) such that σ ⊕ τ and σ′ ⊕ τ′ are unitarily isomorphic.

The set K(F) of stably isomorphic triples is an abelian monoid with respect to to the sum of triples. Denote by d(E, F, α) the class of (E, F, α) in K(F).

Lemma 23. The monoid K(F) is an abelian group. Moreover

d(E, F, α) + d(F, E, α^{∗}) = 0.

Proof. Note that

d(E, F, α) + d(F, E, α^{∗}) = d(E ⊕ F, F ⊕ E, α ⊕ α^{∗})

The last triple is isomorphic to (E ⊕ F, β), where

β = ( \begin{matrix}0 & −α^{∗} \\ α & 0 \end{matrix} )

which is homotopic to 1_{F(E)⊕F(F)} by u(t) = σ(t)\sqrt{σ^{∗}(t)σ(t)}, where

σ(t) = ( \begin{matrix}1 & −tα^{∗} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ tα & 1 \end{matrix} ) ( \begin{matrix}1 & −tα^{∗} \\ 0 & 1 \end{matrix} )

□

The following theorem compares our definition of K(F) with the corresponding one of Karoubi.

Theorem 24. The canonical homomorphism i : K(F) → 𝕂(F) defined by

d(E, F, α) ↦ d(E, F, α)

is an isomorphism.

Proof. Let (E, F, α) be a triple which defines an element in 𝕂(F), where α is an isomorphism (but not unitary isomorphism). Let ᾱ = α\sqrt{α^{∗}α}. ᾱ is unitary and is homotopic to α because α^{∗}α is homotopic to 1_{F(E)}. This proves that i is an epimorphism. Now, let d(E, F, α) ∈ K(F) defines 0 in 𝕂(F). This means by [9] that there exist objects G and H and isomorphisms (but after polar decomposition we may suppose they are unitary isomorphisms) u : E ⊕ G → H and v : F ⊕ G → H that F(v)(α ⊕ 1_{F}(G))F(u^{∗}) is homotopic to 1_{F(H)} (see [9]) by a homotopy h(t). Then h̄(t) = h(t)\sqrt{h^{∗}(t)h(t)} gives homotopy between (E, F, α) ⊕ (G, G, 1_{G}) and (H, H, 1_{H}). This means d(E, F, α) = 0 in K(F). □

This theorem means that all properties of K(F) inherited from the corresponding properties of 𝕂(F). In particular, we get the following results. (Cf. [8], [9].)

Lemma 25. There are the following relations in K(F):

a). If α and α′ are homotopic, then d(E, F, α) = d(E, F, α′);

b). d(E, F, α) + d(F, G, β) = d(E, G, βα).

Proposition 26. Let F : A → B be a Serre quasisurjective additive ∗-functor.

a). If in the definition of K(F) we replace elementary triples by trivial triples then we get the same group.

b). d(E, F, α) = 0 iff there exist an object G from A and unitary isomorphism β : E ⊕ G → F ⊕ G such that F(β) = α ⊕ 1_{F(G)}.

Proposition 27. Let F : A → B be a quasi-surjective additive ∗-functor. Then the sequence of abelian groups

K^{−1}(A) ⟶^{f_{1}} K^{−1}(B) ⟶^{∂} K^{0}(F) ⟶^{i} K^{0}(A) ⟶^{∂} K^{0}(B)  (7)

is exact, where i(d(E, F, α)) = d(E) − d(F) (for the definition of ∂ see [9]). In addition, if there exists a functor Ψ : B → A such that F ·Ψ ≃ Id_{B}, then there exists a split exact sequence

0 → K^{0}(F) →^{i} K^{0}(A) →^{j} K^{0}(B) → 0.  (8)

Example 28. 1)Recall that an object of rep(A, B) has the form (E, ϕ), where E is a right Hilbert B-module and ϕ : A → L(E) is supposed equivariant. A morphism from (E, ϕ) to (E′, ϕ′) is by definition an invariant B-homomorphism f : E → E′ such that fϕ(a) = ϕ′(a)f (see example 11). Note that rep(A, B) is a pseudoabelian C^{∗}-category. To show that K^{i}(rep(A, B)) = 0 for all i ∈ ℤ_{2}, consider the ∞-structure of rep(A, B) E^{∞} = E ⊕ E ⊕ ⋯, α^{∞} = α ⊕ α ⊕ ⋯, and ϕ^{∞}(a) = (ϕ(a))^{∞}. Let

∞ : rep(A, B) → rep(A, B)

be the ∗-functor defined by the formula ∞(E) = E^{∞}, ∞(ϕ) = ϕ^{∞}, and if α is a morphism in rep(A, B), then ∞(α) = α^{∞}. There exists a natural isomorphism id_{rep(A,B)} ⊕ ∞ ≃ ∞. From this it follows that the groups K^{i}(rep(A, B)) (and the cancellation monoid C(rep(A, B)) of isomorphism classes of objects of rep(A, B)) have an automorphism I with property that

id_{K^{i}(rep(A,B))} + I = I.

From this fact it follows that K^{i}(rep(A, B)) = 0 (resp., C(rep(A, B) = 0).

2) Consider the canonical quasi-surjective functor

Θ_{A,B} : rep(A, B) → Cal(A, B).

Applying the exact sequence (7) of K-groups and result of example 1, one gets that the canonical homomorphism

∂ : K^{−1}(Cal(A, B)) → K^{0}(Θ_{A,B})  (9)

is an isomorphism. □

4. Remarks on the definition of KK-groups

In this section we review definitions of Kasparov KK-groups in the form that will be needed for our purposes.

Let A and B be ℤ_{2}-graded C^{∗}-algebras, assuming that A is separable and B is σ-unital. Let A^{+} be obtained from A by adjoining a unit of degree 0. We also assume that all C^{∗}-algebras have fixed compact group actions as above.

4.1. Operatorial homotopy and KK-theory

Let E(A, B) be the set of Kasparov A, B-bimodules. Denote by KK(A, B) the Kasparov group obtained by dividing E(A, B) by the equivalence relation generated by unitary isomorphism and operator homotopy, modulo degenerate bimodules [10], [16], [3].

We need the following elementary properties of KK(A, B):

a) “Cofinality principle”. Let F be a cofinal full additive subcategory of ℍ(B). If in the definition of KK(A, B) the Kasparov A, B-bimodules are replaced by Kasparov A, B-bimodules defined by F, then we get the same group.

b) “Unitization principle”. There exists a split exact sequence

0 → KK(A, B) →^{j} KK(A^{+}, B) ↔^{p^{∗},i^{∗}} KK(ℂ, B) → 0.

Remark. Let s : A → A^{+} be the canonical inclusion, p : A^{+} → ℂ the canonical projection, and i : ℂ → A^{+} the canonical inclusion. One has following split exact sequence

0 → KK(A^{+}, A) →^{s^{∗}} KK(A^{+}, A^{+}) ↔^{i_{∗},p_{∗}} KK(A^{+}, ℂ) → 0.

Consider the element 1−ip in KK(A^{+}, A^{+}). One checks that p_{∗}(1−ip) = 0. From this follows that there exists a unique j ∈ KK(A^{+}, A) such that

js = 1_{A} in KK(A, A) and sj + ip = 1_{A^{+}} in KK(A^{+}, A^{+}).

c) “Unitality principle”. Let A be unital C^{∗}-algebra. Then in the definition of KK(A, B) it is possible to take Kasparov A, B-bimodules of form (E, ϕ, F) where ϕ(1) = 1.

The following definition is motivated by b) and c).

Definition 29. A Kasparov A, B-bimodule e = (E, φ, F) is said to be almost unital if e has the following properties:

[φ(a), F] ∈ K(E), F − F^{∗} ∈ K(E), F^{2} − 1 ∈ K(E)  (10)

for all a ∈ A. An almost unital A, B-bimodule (E, φ, F) is degenerate if

[φ(a), F] = 0, F = F^{∗}, F^{2} = 1  (11)

Denote by E^{+}(A, B) set of almost unital Kasparov A, B-bimodules.

Define the group KK^{+}(A, B) by analogy with the definition of KK(A, B), using almost unital A, B-bimodules. By principles b) and c), KK^{+}(A, B) is essentially isomorphic to KK(A^{+}, B).

Remark. a) Let e = (E, φ, F) be an almost unital A, B-module, and π : L(E) → Cal(E) be a canonical ∗-homomorphism. The morphism π(F) is a unitary morphism in Cal(E), and thus ||π(F)|| = 1. Apply the lifting theorem from [1], which confirms the existence of normpreserving lifting of an element, we get an element G′ ∈ L(E) such that G′ − F ∈ K(E) and ||G′|| = 1. Therefore, (E, φ, G′) is also an almost unital A, B-module with ||G′|| = 1. Replacing G′ by G = \frac{G′+G^{′∗}}{2} , one get the A, B-module e′ = (E, φ, G) with the following properties:

[φ(a), G] ∈ K(E), G = G^{∗}, G^{2} − 1 ∈ K(E), ||G|| ⩽ 1  (12)

and G − F ∈ K(E). This fact implies that e is operatorial homotopy to e′, connected by segment. The A, B-modules with properties 12 will be called fine A, B-modules.

b) Let (E, φ, G_{0}) and (E, φ, G_{1}) be two fine A, B-modules connected by an operatorial homotopy (E, φ, G_{t}) of almost unital A, B-modules. Using the same technic as in a), one gets an operatorial homotopy (E, φ, G′_{t}) of fine A, B-modules. When i = 0, 1, the A, B-modules (E, φ, G_{i}) and (E.φ, G′_{i}) are trivially operatorial homotopic. Thus (E, φ, G_{0}) and (E, φ, G_{1}) are operatorial homotopic in the set of fine A, B-modules.

The above remark shows that KK^{+}(A, B) (resp., E_{+}(A, B), CE^{+}A, B, KE_{+}(A, B)) does not change if one replaces almost unital A, B-modules by the fine A, B-modules in the constructions.

Now, consider G(A, B) the set of A, B-bimodules (E, ϕ, G) with the following properties:

[ϕ(a), G] ∈ K(E), G = G^{∗}, G^{2} = 1.  (13)

which will be called best A, B-modules. Let G(A, B) be the abelian monoid of equivalence classes of A, B-bimodules from G(A, B), equivalence being generated by unitary isomorphism and operatorial homotopy. Let CG(A, B) be the cancellation monoid (resp., KG(A, B) be the Grothendieck group) of G(A, B). The essential map G(A, B) → E(A, B) induces homomorphisms

μ : CG(A, B) → KK^{+}(A, B)

and

α : CG(A, B) → KK^{+}(A, B)

since KK^{+}(A, B) is abelian group. According on the last remark, one has

Proposition 30. Let A and B be as above. Then the following sequence of groups is split exact

0 → CG(A, B) →^{μ} KK^{+}(A, B) ↔^{p^{∗},i^{∗}} KK(ℂ, B) → 0.  (14)

Also, the canonical homomorphisms CG(A, B) → KG(A, B) and

α : CG(A, B) → KK(A, B)

are isomorphisms.

Proof. Firstly we show that μ is a monomorphism. Let (E, φ, G) and (E′, φ′, G′) be best A, B-modules, and let them define the same element of KK^{+}(A, B). Then there exist degenerate A, B-modules (Ê, φ̂, Ĝ) and (Ê′, φ̂′, Ĝ′) such that

(E, φ, G) ⊕ (Ê, φ̂, Ĝ)

is operatorial homotopic in the set of fine A, B-modules to

(E′, φ′, G′) ⊕ (Ê′, φ̂′, Ĝ′).

Let (R, ϕ, L_{t}) be this operatorial homotopy. Since ||L_{t}|| ⩽ 1 for t ∈ [0; 1], one has the following operatorial homotopy in the set of best A, B-modules:

(R ⊕ R, ϕ ⊕ 0, L′_{t})

where

L′_{t} = ( \begin{matrix}L_{t} & (1 − L_{t}^{2})^{1/2} \\ (1 − L_{t}^{2})^{1/2} & −L_{t} \end{matrix} ) ,

which gives operatorial homotopy in the set of best A, B-modules. This means that the elements (E, φ, G) and (E′, φ′, G′) are operatorial homotopic in the set of best A, B-modules up to adding of degenerate A, B-modules. But from the example 28 follows that degenerate A, B-modules define zero element in CG(A, B). Thus (E, φ, G) is equal to (E′, φ′, G′) in CG(A, B). Now we prove that ker(KK^{+}(A, B) → KK(A, B)) ⊂ im(μ). Let (E, φ, G) be fine A, B-module such that the induced ℂ, B-module defines zero element in KK(ℂ, B). Then A, B-module (E, 0, G) defines zero element in KK^{+}(A, B) too. Thus the best A, B-module (E ⊕ E, φ ⊕ 0, D), where

D = ( \begin{matrix}G & (1 − G^{2})^{1/2} \\ (1 − G^{2})^{1/2} & −G \end{matrix} ) ,  (15)

defines the same element as (E, φ, G) in KK^{+}(A, B). To prove the other statements of exactness is trivial and left to the reader. As a consequence we get that CG(A, B) is an abelian group, and thus it is isomorphic to KG(A, B). Finally, comparing our split exact sequence with the exact sequence of ”unitization principle”, one gets that α is an isomorphism. □

4.2. Fredholm picture for the trivially graded case

In this subsection we consider the case when A and B are trivially graded C^{∗}-algebras with compact group actions. Let E = E^{0} ⊕E^{1} be a graded Hilbert B-module. Then E^{0} is a trivially graded B-module and E^{1} = M̌, where M is a trivially graded Hilbert B-module. Consider e = (E, ϕ, F), an almost unital Kasparov (A, B)-bimodule. Then in the decomposition E = E^{(0)} ⊕ E^{(1)}, one has

ϕ = ( \begin{matrix}ϕ^{(0)} & 0 \\ 0 & ϕ^{(1)} \end{matrix} )

and

F = ( \begin{matrix}0 & F^{(0)} \\ F^{(1)} & 0 \end{matrix} )

where (E^{(0)}, ϕ^{(0)}) and (E^{(1)}, ϕ^{(1)}) are A, B-bimodules and F^{(i)} : (E^{(i)}, ϕ^{(i)}) → (E^{(i+1)}, ϕ^{(i+1)}) are bimodule morphisms of degree 1. This interpretation motivates the following:

Definition 31. a). Let A and B be C^{∗}-algebras with compact group actions. A Fredholm A, B-bimodule is μ = (E^{(0)}, ϕ^{(0)}, E^{(1)}, ϕ^{(1)}, F), where E^{(i)}, i = 0, 1, are trivially graded Hilbert B-modules, and (E^{(0)}, ϕ^{(0)}) and (E^{(1)}, ϕ^{(1)}) are A, B-bimodules, and F : (E^{(0)}, ϕ^{(0)}) → (E^{(1)}, ϕ^{(1)}) is a morphism of A, B-bimodules of degree 0, i.e.,

F ϕ^{(0)}(a) − ϕ^{(1)}(a)F ∈ K(E^{(0)}, E^{(1)})

such that F^{∗}F − 1 ∈ K(E^{(0)}) and F F^{∗} − 1 ∈ K(E^{(1)}). The set of Fredholm A, B-bimodules is denoted by F(A, B).

b). Fredholm bimodules

μ = (E^{(0)}, ϕ^{(0)}, E^{(1)}, ϕ^{(1)}, F)

and

μ̄ = (Ē^{(0)}, ϕ̄^{(0)}, Ē^{(1)}, ϕ̄^{(1)}, F̄)

are called unitarily isomorphic if there exist unitary B-isomorphisms u : E^{(0)} → Ē^{(0)} and v : E^{(1)} → Ē^{(1)} such that uϕ^{(0)}u^{∗} = ϕ̄(0), vϕ^{(1)}v^{∗} = ϕ̄^{(1)} and uF v^{∗} = F̄.

c). An operator homotopy through Fredholm bimodules is

μ_{t} = (E^{(0)}, ϕ^{(0)}, E^{(1)}, ϕ^{(1)}, F_{t})

which is a Fredholm bimodule for all t ∈ [0, 1], such that t ↦ F_{t} is norm continuous. d). The addition of Fredholm bimodules is defined by the formula

μ ⊕ μ̄ = (E^{(0)} ⊕ Ē^{(0)}, ϕ^{(0)} ⊕ ϕ̄^{(0)}, E^{(1)} ⊕ Ē^{(1)}, ϕ^{(1)} ⊕ ϕ̄^{(1)}, F ⊕ F̄)

e). Let F (A, B) be the monoid of equivalence classes of Fredholm bimodules, where equivalence is generated by unitary isomorphism and operator homotopy, and let CF (A, B) (resp., KF (A, B)) be its cancellation semigroup (resp., Grothendieck group).

Now our concern is to compare the notion of Fredholm bimodule and Kasparov bimodule.

Construction A Let μ = (E^{(0)}, ϕ^{(0)}, E^{(1)}, ϕ^{(1)}, F) be a Fredholm A, B-bimodule and Ě^{(1)} be the opposite

graded Hilbert B-module of E^{(1)}. Let ϕ̌^{(1)} : A → L(Ě^{(1)}) be the opposite to ϕ^{1}. Define an almost unital Kasparov A, B-bimodule μ̄ as the triple (Ē, ϕ̄, F̄), where

Ē = E^{(0)} ⊕ Ě^{(1)}, ϕ̄ = ( \begin{matrix}ϕ^{(0)} & 0 \\ 0 & ϕ̌^{(1)} \end{matrix} ) , F̄ = ( \begin{matrix}0 & F̆^{∗} \\ F̆ & 0 \end{matrix} )

and F̆ is the composition of F with the canonical B-homomorphism

τ_{E^{(1)}} : E^{(1)} → Ě^{(1)}

of degree 1 (see example 7).

Proposition 32. Let χ : F(A, B) → E^{+}(A, B) be defined by μ ↦ μ̄. Then the induced homomorphism of semigroups χ : F (A, B) → E^{+}(A, B) is an isomorphism. Therefore the groups CF (A, B), KF (A, B) and KK^{+}(A, B) are canonically isomorphic.

Proof. The first part is easy to check and the second part follows using proposition 30. □ Now consider the canonical quasi-surjective additive ∗-functor

Θ : Rep(A, B) → Cal(A, B).

We can define Karoubi’s group K(Θ). There exists a canonical homomorphism η : CF → K(Θ) which maps the Fredholm A, B-bimodule

μ = (E^{(0)}, ϕ^{(0)}, E^{(1)}, ϕ^{(1)}, F)  (16)

to

η(μ) = (E^{(0)}, πϕ^{(0)}, E^{(1)}, πϕ^{(1)}, π(F)).

(See example 11 for the definition of π.)

Proposition 33. The homomorphism η : CF (A, B) → K^{0}(Θ_{A,B}) is an isomorphism.

Proof. It is enough to show that η is a monomorphism. Let (16) be a Fredholm bimodule such that η(μ) = 0. Then there exist elementary Fredholm bimodules ε = (E, ϕ, E, ϕ, H) and ω = (Ē, ϕ̄, Ē, ϕ̄, H̄) and an operator homotopy μ ⊕ ε ≃ ω. But the cancellation monoid of the abelian monoid of isomorphic elementary Fredholm bimodules is 0 (cf. example 28). Thus μ = 0 in CF (A, B). □

Combining the above propositions we have the following:

Corollary 34. Let A and B be as above. Then the canonical homomorphism

ηχ^{−1} : KK^{+}(A, B) → K(Θ_{A,B})

is an isomorphism.

4.3. An interpretation of KK^{1}(A, B)

In this subsection we assume that all C^{∗}-algebras are trivially graded and equipped with a compact group action. We use the property of the subcategory of ℍ(B ⊗ C^{(1,0)}), given in subsection 2.4, for characterization of KK^{1}(A, B).

In detail, using the “cofinality principle” of subsection 4.1, and according to the proposition 14, one can use the category ℍ_{∗}(B) for the definition of KK^{1}(A, B). Firstly note that if E ∈ H(B) then E ⊕ Ě ∈ ℍ_{∗}(B). Denote by L_{∗}(E, E′) the space of morphisms from E ⊕ Ě to E′ ⊕ Ě′ in the category ℍ_{∗}(B).

Consider G_{∗}(A, B) the set of triples (E ⊕ Ě, ϕ, g), where E ∈ H(B),

ϕ : A → L_{∗}(E ⊕ Ě)

is a ∗-homomorphism, and g ∈ L_{∗}(E ⊕ Ě), with

[g, ϕ(a)] ∈ L_{∗}(E ⊕ Ě), g^{∗} = g, g^{2} = 1, ∂(g) = 1.

Thus using triples from G_{∗}(A, B) and the analogue of the construction of KK(A, B), one gets KK^{1}(A, B). (cf. [2])

Denote by G_{∗}(A, B) the abelian monoid of equivalence classes of G(A, B), where equivalence is generated by operator homotopy and unitary isomorphism. From the properties of the category ℍ_{∗}(A, B) it follows that

ϕ = ( \begin{matrix}ϕ_{0} & 0 \\ 0 & ϕ̌_{0} \end{matrix} )  (17)

because all elements of A have degree 0, and

g = ( \begin{matrix}0 & ǧ_{0} \\ g_{0} & 0 \end{matrix} )  (18)

(as g has degree 1). From g^{∗} = g and g^{2} = 1 follows that g_{0}^{∗} = ǧ_{0}. Thus ḡ = τĚg_{0} has degree 0, and ḡ^{∗} = ḡ, ḡ^{2} = 1.(see example 7). From this construction follows that one can consider G(A, B) as the set of triples of form (E, ϕ, g) where E is trivially graded Hilbert B-module ϕ : A → L(B) is equivariant ∗-homomorphism and g ∈ L(B) such that

[ϕ(a), g] ∈ K(E), g^{∗} = g, g^{2} = 1 and ∂g = 0, where a ∈ A.

We conclude that canonical homomorphism ρ : CG(A, B) → KK^{1}(A, B) defined by

(E, ϕ, g) ↦ (E ⊕ Ě, ( \begin{matrix}ϕ & 0 \\ 0 & ϕ̌ \end{matrix} ) , ( \begin{matrix}0 & gτĚ \\ −τ_{E}g & 0 \end{matrix} ))

is an isomorphism.

Construction B Denote by P(A, B) the set of triples of the form (E, ϕ, p), where E is a trivially graded

B-module, ϕ : A → L(E) is a ∗-homomorphism, ∂(p) = 0 and

\begin{matrix}[ϕ(a), p] ∈ K(E), a ∈ A; \\ p^{∗} = p and p^{2} = p \end{matrix}

Define a map ϑ : P(A, B) → G_{∗}(A, B) as follows. For each triple (E, ϕ, p) consider the triple (E, ϕ, g_{p}), where g_{p} = (2p − 1). □

Let P (A, B) be the abelian monoid of equivalence class of bimodules of the above form, where equivalence is generated by unitary isomorphism and operator homotopy. Let CP (A, B) (resp,. KP (A, B)) be the cancellation monoid (resp., Grothedieck group) of P (A, B).

Lemma 35. The map ϑ : P(A, B) → G_{∗}(A, B), defined in construction B, is a bijection.

Proof. The inverse map is defined by the formula:

(E, ϕ, g) ↦ (E, ϕ, p_{g}),

where p_{g} = \frac{1−g}{2} □.

Thus we have

Theorem 36. Let A be separable and B be σ-unital trivially graded C^{∗}-algebras with compact group action. Then the canonical homomorphism

θ : KP (A, B) → KK^{1}(A, B),  (19)

defined as the composition θ = ρϑ, is an isomorphism.

5. Main Theorem

In this section we prove the following main result.

Theorem 37. Let A be separable and B be σ-unital trivially graded C^{∗}-algebras with compact group action. There exists an essential isomorphism

κ : K^{j}(Rep(A, B)) ⟶^{≃} KK^{j+1}(A, B),  (20)

where Rep(A, B) is the pseudoabelian C^{∗}-category associated with the additive C^{∗}-category Rep(A, B) and j = −1, 0.

The proof of this theorem consists of two parts, considered in subsections 5.1 and 5.2.

5.1. Proof of theorem in dimension zero

Firstly, recall the definition of K^{0}(Rep(A, B)). By definition 12, objects of Rep(A, B) have the form (E, ϕ, p), where E is a Hilbert B-module, ϕ : A → L(E) is a ∗-homomorphism and p is a projection, (i.e., p^{2} = p and p^{∗} = p) such that

ϕ(a)p − pϕ(a) ∈ K(E).

Two objects (E, ϕ, p) and (F, ψ, q) are unitarily isomorphic in the C^{∗}-category Rep(A, B) (or C^{∗}-categorically unitarily isomorphic) if there exists a partial isometry v : E → F such that

vϕ(a) − ψ(a)v ∈ K(E, F)

for all a ∈ A, v^{∗}v = p and vv^{∗} = q. Define by K^{0}(Rep(A, B)) the Grothendieck group of the abelian monoid of unitarily isomorphic objects. This coincides with Karoubi’s analogous definition (see section 2). On the other hand we may define w-unitary isomorphism of objects: (E, ϕ, p) and (F, ψ, q) are w-unitary isomorphic if there exists a unitary u : E → F such that

uϕ(a) − ψ(a)u ∈ K(E, F)

for all a ∈ A, up = qu. Denote by K_{w}^{0}(Rep(A, B)) the Grothendieck group of the abelian monoid of w-unitary isomorphic objects. Let us compare these groups. If two objects (E, ϕ, p) and (F, ψ, q) are w-unitarily isomorphic by a unitary u, then qup is an isomorphism between these objects. Conversely, if v : (E, ϕ, p) → (F, ψ, q) is a unitary isomorphism of the given objects, then v : E → F is a partial isometry with v^{∗}v = p and vv^{∗} = q. Consider the objects (E, ϕ, 1) and (F, ψ, 1). Then

ω : (E, ϕ, p) ⊕ (F, ψ, 1) → (F, ψ, q) ⊕ (E, ϕ, 1)

is a w-isomorphism, where

ω = ( \begin{matrix}v & 1 − q \\ 1 − p & v^{∗} \end{matrix} ) .  (21)

¿From this remark it follows that

(E, ϕ, p) ⊕ (F, ψ, 1) and (F, ψ, q) ⊕ (E, ϕ, 1)

are equal in K_{w}^{0}(Rep(A, B)), Note that K^{0}(rep(A, B)) = 0, by example 28. From this fact it follows that

(E, ϕ, 1) = (F, ψ, 1) = 0

in K_{w}^{0}(Rep(A, B)). Thus

(E, ϕ, p) = (F, ψ, q)

in the last group. Therefore, there is a correctly defined essential isomorphism

ξ : K^{0}(Rep(A, B)) → K_{w}^{0}(Rep(A, B)).  (22)

The next step is to define a homomorphism

μ : K_{w}^{0}(Rep(A, B)) → KP (A, B).  (23)

To do this one needs the following:

Lemma 38. Let (E, ϕ, p) and (E, ψ, p) be two objects such that

ϕ(a) − ψ(a) ∈ K(E)

for all a ∈ A. Then (E, ϕ, p) ⊕ (E, ψ, 1) and (E, ϕ, 1) ⊕ (E, ψ, p) are operator homotopic.

Proof. Consider

p_{t} = ( \begin{matrix}1 − cos^{2} t · (1 − p) & sin t · cos t · (1 − p) \\ sin t · cos t · (p − 1) & 1 + sin^{2} t · (p − 1) \end{matrix} ) .

Then (E ⊕ E, ϕ ⊕ ψ, p_{t}) is desired operator homotopy.

¿From this lemma follows that the map (E, ϕ, p) ↦ (E, ϕ, p) correctly defines the epimorphism μ and also the epimorphism

μ · ξ : K^{0}(Rep(A, B)) → KP (A, B).

To prove that μ · ξ is a monomorphism one needs the following:

Lemma 39. Let (E, ϕ, p_{0}) and (E, ϕ, p_{1}) be operator homotopic triples. Then they are unitarily isomorphic as objects in Rep(A, B).

Proof. Let (E, ϕ, p_{t}) be an operator homotopy. As [0, 1] is a compact space, one can choose finite set of points t_{0}, t_{1}, …, t_{n} ∈ [0, 1] such that ∥p_{t_{i+1}} − p_{t_{i}}∥ < 1 for all i = 0, …, n − 1. Using lemma 6.4 of [10], one gets that p_{t_{i+1}} and p_{t_{i}} are unitarily isomorphic in the C^{∗}-algebra of morphisms from (E, ϕ) to (E, ϕ) (in the category Rep(A, B)). From this it follows that the triples from the lemma are isomorphic objects in Rep(A, B). □

Now one can define a canonical isomorphism

κ : K^{0}(Rep(A, B)) → KK^{1}(A, B)  (24)

as the composition of isomorphisms ξ, μ and isomorphism θ of 19.

5.2. Proof of theorem in dimension one Consider the quasi-surjective Serre additive ∗-functor

Π_{A,B} : Rep(A, B) → Cal(A, B).

Applying the exact sequence (7), one gets the following exact sequence:

K^{−1}(Rep(A, B)) →^{Π_{∗}} K^{−1}(Cal(A, B)) →^{∂_{Π}} K^{0}(Π) →^{i} K^{0}(Rep(A, B)).

The properties of this sequence are given below.

Lemma 40. There exists a canonical isomorphism

τ : K^{0}(Π_{A,B}) → K^{0}(Θ_{O,B}),  (25)

where O is the zero C^{∗}-algebra.

Proof. Consider e = (E, ϕ, E′, ψ, α) such that α\overline{ϕ(a)} = \overline{ψ(a)}α for all a ∈ A, where \overline{ϕ(a)} = πϕ(a), \overline{ψ(a)} = πψ(a). Then e defines an element in K^{0}(Π_{A,B}). Let 0_{ϕ} : O → L(E) and 0_{ψ} : O → L(E′) be the zero homomorphisms. One has e_{0} = (E, 0_{ϕ}, E′, 0_{ψ}, α) that gives an element in K^{0}(Θ_{O,B}). Conversely, let e_{0} = (E, 0_{ϕ}, E′, 0_{ψ}, α) define an element from K^{0}(Θ_{O,B}). Then the corresponding element in K^{0}(Π_{A,B}) is defined by the same e_{0}, with 0_{ϕ} and 0_{ψ} the trivial homomorphisms on A. Thus we have two homomorphisms

τ : K^{0}(Π_{A,B}) → K^{0}(Θ_{0,B})

and

η : K^{0}(Θ_{0,B}) → K^{0}(Π_{A,B})

such that τη = 1. To prove that τ is an isomorphism, it is enough to show that {e}−{e_{0}} = 0. In order to show this, consider

(E ⊕ E′, ϕ ⊕ 0_{ψ}, E′ ⊕ E, ψ ⊕ 0_{ϕ}, α ⊕ α^{∗}).

It is isomorphic to

(E ⊕ E′, ϕ ⊕ 0_{ψ}, E ⊕ E′, 0_{ϕ} ⊕ ψ, t(α ⊕ α^{∗}),

where

t(α ⊕ α^{∗}) = ( \begin{matrix}0 & α^{∗} \\ α & 0 \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ( \begin{matrix}0 & α^{∗} \\ −α & 0 \end{matrix} ) .  (26)

Note that ( \begin{matrix}0 & α^{∗} \\ −α & 0 \end{matrix} ) is homotopic to ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ). Thus α ⊕ α^{∗} lifts to an isomorphism from

ϕ ⊕ 0_{ψ} to ψ ⊕ 0_{ϕ}. That concludes the proof of the lemma (see subsection 3.2). □

Lemma 41. Let e = (E, ϕ, α) and e′ = (E, ϕ, β) both define elements in K^{−1}(Rep(A, B)). If α − β ∈ K(E), then {e} = {e′}.

Proof. Consider the triples (Ē, ϕ̄, ᾱ) and (Ē, ϕ̄, β̄) where

Ē = E ⊕ E, ϕ̄ = ( \begin{matrix}ϕ & 0 \\ 0 & 0 \end{matrix} ) , ᾱ = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) , β̄ = ( \begin{matrix}β & 0 \\ 0 & α \end{matrix} )  (27)

These triples are operator homotopic by

h(t) = ( \begin{matrix}α + sin^{2} t · (β − α) & sin t · cos t · (α − β) \\ sin t · cos t · (β − α) & β + sin^{2} t · (α − β) \end{matrix} ) . □

Lemma 42. The canonical homomorphism

Π_{∗} : K^{−1}(Rep(A, B)) → K^{−1}(Cal(A, B))

is a monomorphism.

Proof. Let e = (E, ϕ, α) represent an element in K^{−1}(Rep(A, B)) such that Π_{∗}(e) = 0. Then by lemma 20 there exists (E′, ψ, 1_{ψ}) such that Π(α) ⊕ 1_{Π(ψ)} is homotopic to 1_{Π(ϕ)⊕Π(ψ)}. Let h(t) be this homotopy. Consider its lifting in the group U(E ⊕ E′, ϕ ⊕ ψ) of unitary automophisms, such that h(0) = 1. Put β = h(1), then β − α ⊕ 1_{ψ} ∈ K(E). Using lemma 18 and lemma 41, we get

{(E, ϕ, α) ⊕ (F, ψ, 1)} = {E ⊕ F, ϕ ⊕ ψ, β} = 0. □

Let e = {(E, ϕ, E′, ψ, α)} ∈ K^{0}(Π_{A,B}). Consider the unique element e′ = {(0, β)} ∈ K^{−1}(Cal(A, B)) such that ∂_{Π}(e′) = e. Then the homomorphism

s : K^{0}(Π) → K^{−1}(Cal(A, B))

defined by e ↦ e′defines a right inverse for ∂_{Π}.

Corollary 43. Let A and B be trivial graded C^{∗}-algebras with compact group actions. Then there is a split exact sequence of groups

0 → K^{−1}(Rep(A, B)) ⟶^{Π} K^{−1}(Cal(A, B)) ⟶^{τ·∂_{Π}} K^{0}(Θ_{O,B}) → 0.  (28)

Now we are ready to prove the main theorem. Firstly note that one can replace K^{−1}(Rep(A, B)) by K^{−1}(Rep(A, B)) (see theorem 21).

We need some homomorphisms:

a) Define the homomorphism

κ : K^{−1}(Rep(A, B)) → KK^{0}(A, B)

as follows. Let e = (E, ϕ, α) define an element of K^{−1}(Rep(A, B)). Consider the triple ē = (Ē, ϕ̄, ᾱ), defined in construction A of section 3, by

Ē = E ⊕ Ě, ϕ̄ = ( \begin{matrix}ϕ & 0 \\ 0 & ϕ̌ \end{matrix} ) , ᾱ = ( \begin{matrix}0 & α̌ \\ α & 0 \end{matrix} ) .

Then by definition κ(e) = ē.

b) Define homomorphism

Δ : K^{−1}(Cal(A, B)) → KK^{+}(A, B)

in the following way. Let e = (E, πϕ, α) be a triple that defines element of K^{−1}(Cal(A, B)), where (E, ϕ) is object of Rep(A, B). Choose F such that π(F) = α. Of course, the triple e′ = (E, ϕ, F) is an almost unital Kasparov A, B-bimodule. By definition Δ(e) = ē′.

c) Let ω : K(Θ_{O,B}) → KK^{0}(ℂ, B) be defined by the equality ω = χη^{−1}.

Consider the commutative diagram

0 → K^{−1}(Rep(A, B)) →^{Π} K^{−1}(Cal(A, B)) →^{τ·∂} K^{0}(Θ_{O,B}) → 0
↓^{κ}  ↓^{Δ}  ↓^{ω}
0 →  KK^{0}(A, B)  →^{j} KK^{+}(A, B) →^{i^{∗}} KK^{0}(ℂ, B) → 0

Then Δ and ω are isomorphisms, because Δ = χη^{−1}∂_{Θ} is the composition of the isomorphism from corollary 34 and the isomorphism (9) of example 28. Also, ω = χη^{−1} is an isomorphism because of lemma 40. Thus κ is an isomorphism.

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Tamaz Kandelaki kandel@rmi.acnet.ge

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia