In [3], Cuntz noted that if F is a homotopy invariant, stable and split exact functor from the category C^* of separable C^*-algebras to the category of abelian groups then Kasparov's KK-theory acts on F, that is, every element of KK(A,B) induces a natural map F(A) ⇒ F(B). Higson [4], on the other hand, developed Cuntz' findings further and proved that every such functor F factorizes through the category K consisting of separable C^*-algebras as the object class and KK-theory together with the Kasparov product as the morphism class, that is, F is the composition F^∘κ of a universal functor κ from the class of C*-algebras to K and a functor F^ from K to abelian groups.

In [12], Thomsen generalized Higson's findings to the group equivariant setting by replacing everywhere in the above statement algebras by equivariant algebras, *-homomorphisms by equivariant *-homomorphisms and KK-theory by equivariant KK-theory (the proof is however far from such a straightforward replacement). Meyer [9] used a different approach in generalizing Higson's result, and generalized it to the setting of action groupoids G⋉X.

In this note we extend Higson's universality result to the inverse semigroup equivariant setting.

More specifically, the following will be shown. Let G be a countable unital inverse semigroup and denote by C* the category consisting of (ungraded) separable G-equivariant C^*-algebras as objects and G-equivariant *-homomorphisms as morphisms. Denote by Ab the category of abelian groups. A functor F from C* to Ab is called stable if F(φ) is an isomorphism for every G-equivariant corner embedding φ:A→A⊗K, where A is an G-C^*-algebra, K denotes the compacts on some separable Hilbert space, and A⊗K is a G-algebra where the G-action may be arbitrary and not necessarily diagonal. Similarly, F is called homotopy invariant if any two homotopic morphisms φ0,φ1:A→B in C* satisfy F(φ0)=F(φ1), and split exact if F turns every short split exact sequence in C* canonically into a short split exact sequence in Ab.

Let KG the denote the category consisting of separable G-equivariant C^*-algebras as object class and the G-equivariant KK-theory group KK^G(A,B) as the morphism set between two objects A and B; thereby, composition of morphisms is given by the Kasparov product (g°f:=f⊗Bg for f∈KKG(A,B) and g∈KKG(B,C)). The one-element of the ring KK^G(A,A) is denoted by 1_A.

K^G is an additive category A, that means, the homomorphism set A(A,B) is an abelian group and the composition law is bilinear. We call a functor F from C^* into an additive categtory A homotopy invariant, stable and split exact if the functor B↦A(F(A),F(B)) enjoys these properties for every object A. This notion is justified by the fact that this property is equivalent to saying that F itself satisfies these properties in the sense introduced above for Ab, see the properties (i)-(iii) in Higson [4], page 269, or Lemma 12.2.

Let κ denote the functor from C^* to K^G which is identical on objects and maps a morphism g:A→B to the morphism g*(1A)∈KKG(A,B). Then we have the following proposition, theorems and corollary.

Proposition 1.1.

Let G be a countable unital inverse semigroup. Let A be an object in C^*. Then B↦KKG(A,B) is a homotopy invariant, stable and split exact covariant functor from C^* to Ab.

Theorem 1.2.

Let G be a countable unital inverse semigroup. Let F be a homotopy invariant, stable and split exact covariant functor from C^* to Ab. Let A be an object in C^* and d an element in F(A). Then there exists a unique natural transformation ξ from the functor B↦KKG(A,B) to the functor F such that ξA(1A)=d.

Theorem 1.3.

Let G be a countable unital inverse semigroup. Let F be a homotopy invariant, stable and split exact covariant functor from C^* to an additive category A. Then there exists a unique functorF^ from K^G to A such that F=F^°κ, where κ:C*→KG denotes the canonical functor.

Corollary 1.4.

The above results are also valid for countable non-unital inverse semigroups G and for countable discrete groupoids G.

A very brief overview of this note is as follows. (We give further short summaries at the beginning of each section).

In Section 2 we briefly recall the definitions of inverse semigroup equivariant KK-theory. In Section 4 we prove Proposition 1.1. In Section 8 we establish a Cuntz picture of KK-theory. The core of how to associate homomorphisms in the image of a homotopy invariant, stable and split-exact functor F to Kasparov elements is explained in Section 9. Finally Theorem 1.2, Theorem 1.3 and Corollary 1.4 are proved in Sections 11, 12 and 13, respectively.

Sections 4--11 present an adaption of the content of Thomsen's paper [12]; this goes mostly without saying. Section 12 is essentially taken from Higson's paper [4].

We finally would like to clarify the implications of our results to groupoid equivariant KK-theory. Note that there exists a natural transition between inverse semigroups and groupoids by Paterson [10]. Usually these groupoids are non-Hausdorff topological spaces, and thus our results cover a subclass of non-Hausdorff topological groupoid equivariant KK-theory. The equivariant KK-theory of topological, not necessarily Hausdorff groupoids is defined in Le Gall [8]. The G-actions used in this paper, see Def. 2.2, correspond exactly to the G-actions used by Le Gall, and are based on C₀(X)-algebras, which go back to Kasparov and his C₀(X)-G-equivariant KK-theory RKKG in [7]. See also [2] for more details on this translation.

Actually, all our results of this paper work for all topological, not necessarily Hausdorff groupoids equivariant KK-theory groups by minor and easy adption.

Indeed, let G be a locally compact groupoid with base space X. At first we may consider it as a discrete inverse semigroup S by adjoining a zero element to G, i.e. set S:=G∪{0}.

A G-action α on A is then fiber-wise just like an inverse semigroup S-action on A (the zero element 0 ∈ S acts always as zero), with the additional property that it is continuous in the sense that it forms a map α:s*A→r*A. We cannot, as in inverse semigroup theory, say that αss−1(A) is a subalgebra of A (s ∈ S), because this instead we would interpret as a fiber Ass−1 of A. But all computations done for inverse semigroups would be the same if we did it for a groupoid on fibers. That is why we need only take care that every introduced G-action is continuous.

But the introduced actions, or similar constructions are just:

• (1) Cocycles: The Definition 5.1 has to be replaced by the analogous Definition 1.5 below.

• (2) Unitization: One replaces Definition 3.3 by the corresponding definition of [8].

• (3) Direct sum, internal, external tensor product: It is clear that these constructions are also continuous for groupoids.

• (4) For an element [T,E]∈KKG(A,B) one has the condition that the bundle g↦g(Ts(g))−Tr(g)∈K(Er(g)) is in r*K(E). Here one has also additionally to check continuity.

• (5) One has also to consider C₀(X)-structure of the base space X of the groupoid, which recall is homomorphims C0(X)→ZM(A) (center of the multiplier algebra of A). Any possible necessary computations one makes analogously as we do it for the restricted G-action E→ZM(A) to the idempotent elements E of the inverse semigroup by replacing e ∈ E by f∈C0(X).

Definition 1.5.

Let (A,α) be a (G)-algebra. Set the (G)-action on LA(A)≅M(A) to be α¯:=Ad(α). An α-cocycle is a unitary u in r*(M(A)) such that

in M(A)r(g) for all g,h∈G with s(g)=r(h).

In this way it is (almost) clear that the results of this paper hold also in the locally compact, not necessarily Hausdorff groupoid equivariant setting.

Corollary 1.6.

Let (G) be a locally compact, not necessarily Hausdorff groupoid. Then KK(G) is the universal stable, homotopy invariant and split exact category deduced from the category of (G)-equivariant, separable, ungraded C*-algebras.

Our reference for inverse semigroup equivariant KK-theory is [1]. We shall however exclusively work with its slightly adapted variant called compatible K-theory, as in [2], but denote it by KK^G rather than KKG^ as in [2]. All (exterior) tensor products are however ordinary as in [1] and are not forced to be C₀(X)-balanced as in [2]! (The internal tensor products are automatically C₀(X)-balanced.) For convenience of the reader we completely recall the basic definitions.

Definition 2.1.

Let G denote a countable unital inverse semigroup. The involution in G is denoted by g↦g−1 (determined by gg−1g=g and g−1gg−1=g−1). A semigroup homomorphism is said to be unital if it preserves the identity 1 ∈ G. To include also semigroups with a zero element, we insist that a semigroup homomorphism preserves also the zero element 0 ∈ G if it exists.

We denote the set of idempotent elements of G by E.

Definition 2.2.

A G-algebra (A,α) is a ℤ/2-graded C^*-algebra A with a unital semigroup homomorphism α:G→End(A) such that α_g respects the grading and αgg−1(x)y=xαgg−1(y) for all x,y ∈ A and g ∈ G.

Throughout we shall identify the C^*-algebras LA(A) (adjoint-able operators on the Hilbert A-module A) and M(A) (multiplier algebra of A) by a well-known *-isomorphism. We denote the set of idempotent elements of G by E.

Definition 2.3.

A G-Hilbert B-module E is a ℤ/2-graded Hilbert module over a G-algebra (B,β) endowed with a unital semigroup homomorphism G→Lin(E) (linear maps on E) such that U_g respects the grading and

• (a) ⟨Ug(ξ),Ug(η)⟩=βg(⟨ξ,η⟩)

• (b) U_g(ξ b) = U_g(ξ) β_g(b)

• (c) Ugg−1(ξ)b=ξβgg−1(b)

for all g∈G,ξ,η∈E and b∈B.

Lemma 2.4.

In the last definition, automatically Ugg−1 is a self-adjoint projection in the center of the algebra L(E).

For a positive approximate unit (bi)⊆B we compute

so that Ugg−1 is seen to be self-adjoint. This operator is in the center because

for all T∈L(E),ξ∈E and b∈B.

Definition 2.5.

Given a G-Hilbert B-module (E,U) we turn the C^*-algebra (L)B(E) to a G-algebra under the action g(T):=UgTUg−1 for all g∈G and T∈L(E).

It is useful to notice that every G-algebra (A,α) is a G-Hilbert module over itself under the inner product ⟨a,b⟩=a*b; so we have all the identities of Definition 2.3 for U:=β:=α. Actually Definitions 2.2 and 2.3 are equivalent for C*-algebras. Hence we note that

Lemma 2.6.

Let (A,α) be a G-algebra. Every αe∈(L)A(A)=M(A) for e ∈ E is a self-adjoint projection in the center of M(A). The application of α_e is given by multiplication, that is, αe(a)=aαe in M(A).

Definition 2.7.

A *-homomorphism φ:(A,α)→(B,β) between G-algebras if φ(αg(a))=βg(φ(a)) for all a∈A,g∈G.

Definition 2.8.

A G-Hilbert A,B-bimodule over G-algebras A and B is a G-Hilbert B-module E equipped with a G-equivariant *-homomorphism π:A→L(E) (the left module multiplication operator).

Definition 2.9.

Let A and B be G-algebras. We define a G-equivariant Kasparov A,B-cycle to be an ordinary Kasparov cycle (E,T) without G-action (see [6, 7]) such that however E is a G-Hilbert A,B-bimodule and the operator T∈L(E) satisfies

for all g ∈ G. The Kasparov group KK^G(A,B) is defined to be the collection of all G-equivariant Kasparov A,B-cycles divided by homotopy induced by G-equivariant Kasparov A,B[0,1]-cycles. (Throughout, B[0,1]:=B⊗C([0,1]).)

We equip the multiplier algebra of a G-algebra with a G-action as described in Definition 2.5. This is also the continuous extension of the G-action on A to M(A) in the strict topology. We redundantly emphasize this again:

Definition 2.10.

Given a G-algebra (A,α), LA(A)=M(A) becomes a G-algebra under the G-action α¯:G→End(LA(A)) given by αg¯(T):=αg∘T∘αg−1 for all g ∈ G and T∈LA(A).

We also write α¯:M(A)→M(B) for the strictly continuous extension of a *-homomomorphism α:A→B of C*-algebras.

Definition 2.11.

Write K for the compact operators on a separable Hilbert space. Call a G-algebra (B,β) stable if there is a G-equivariant *-isomorphism (B,β)→(B⊗K,β⊗trivial).

Since every G-algebra can be stabilized, we use the last definition as a convenient way to avoid the cumbersome notation B⊗K.

Definition 2.12.

Let (E,U) be a G-Hilbert A-module. An operator T in L(E) is called G-invariant if T commutes with the operator Ug:E→E (that is, T∘Ug=Ug∘T) for all g ∈ G. (Equivalently: UgTUg−1=TUgg−1.)

Note that then T* automatically commutes also with U_g.

Lemma 2.13.

Let (B,β) be a stable G-algebra. Then there exist G-invariant isometries V₁ and V₂ in LB(B)=M(B) such that V1V1*+V2V2*=1.

Proof. Let K be the compact operators on l2(ℕ). Write ei,j∈K for the standard matrix units. We define the operators V1,V2∈LB⊗K(B⊗K) for example by

for all i,j∈ℕ, b ∈ B and k=1,2.

Lemma 2.14.

Let (B,β) be stable. A G-invariant unitary U∈M(B) can be connected to 1∈M(B) by a G-invariant, strictly continuous unitary path in M(B).

Proof. In the non-equivariant case this is for example [5, Lemma 1.3.7]. Its canonical proof works equivariantly without modification.

Let us point out that we have all the necessary techniques for inverse semigroup equivariant KK-theory that we shall need available: KK^G allows an associative Kasparov product, and KK^G(A,B) is functorial in A and B (see [1]).

From now on all C^*-algebras are assumed to be trivially graded and separable!

We shall later need a unitization of a G-algebra. To this end we cannot simply add a single unit but need to adjoin the whole G-algebra C*(E) to a given G-algebra. This section is dedicated to describe this.

Definition 3.1.

Let C*(E) denote the universal abelian C*-algebra generated by the free set E of commuting self-adjoint projections. This algebra is endowed with the G-action τ induced by τg(e):=geg−1∈E for g∈G,e∈E which turns it to a G-algebra.

Lemma 3.2.

Let (A,α) be a G-algebra. Given a ∈ A and z∈C*(E) write

where γ:C*(E)→M(A) denotes the canonical *-homomorphism such that γe(a)=αe(a) for all e∈E,a∈A.

Then the linear direct sum A⊕C*(E) turns to a G-algebra under the operations

for all a,b∈A,z,w∈C*(E) and under the diagonal G-action α⊕τ.

Proof. In this proof, ⊕ indicates only a linear sum, and ⊕(C*) a C*-direct sum. Put Z:=span(E)⊆C*(E). Of course, Z is a dense *-subalgebra of C*(E).

We leave it to the reader to show that A⊕C*(E) is a *-algebra. We claim that A⊕Z⊆A⊕C*(E) can be equipped with a C*-norm. Given a finite subset F⊆E, write ZF⊆Z⊆C*(E) for the finite-dimensional *-subalgebra of C*(E) generated by F. It is sufficient to define a C*-norm for each single A⊕ZF, since A⊕Z is the directed union over all such *-algebras A⊕ZF, and each direct sum A⊕ZF must then be topologically closed as Z_F is finite-dimensional.

Since Z_F is a finite-dimensional commutative C*-algebra, there is an isomorphism ψ: ℂn→ZF of C*-algebras. Assume by induction hypothesis for 0≤k<n that A⊕ψ(ℂk)⊆A⊕C*(E) is a C*-algebra. Let z=ψ(ek+1), where ek+1=0⊕⋯⊕0⊕1ℂ∈ ℂk+1. Multiplication in A⊕ψ(ℂk+1) is as follows:

for x,y∈A⊕ψ(ℂk)⊆A⊕ψ(ℂk+1) and λ,μ∈ℂ. If γz∈A then

defines a *-isomorphism (x∈A⊕ψ(ℂk),λ∈ℂ),), whence A⊕ψ(ℂk+1) is a C*-algebra.

If γz∉A then consider the operator P∈M(A⊕ψ(ℂk)) defined by P(x)=x z. Observe that P∉A⊕ ℂk (as otherwise P and so γ_z would be in A, since ψ(ℂk)z=0). The injective *-homomorphism

(x∈A⊕ψ(ℂk),λ∈ ℂ) shows that A⊕ψ(ℂk+1) is a C*-algebra. This completes induction. Thus A⊕ψ(ℂn)≅A⊕ZF is a C*-algebra.

Note that the canonical projection A⊕ZF→ZF⊆C*(E) is a *-homomorphism of C*-algebras and thus contractive. Thus by taking the direct limit we obtain a canonical contractive projection A⊕Z¯→C*(E). Since A⊕C*(E)⊆A⊕Z¯, we have a contractive projection A⊕C*(E)→C*(E). By a standard result in the theory of topological vector spaces, A⊕C*(E) is complete. Thus A⊕C*(E)=A⊕Z¯ is a C*-algebra.

Finally, a straightforward check shows that A⊕C*(E) is a G-algebra.

Definition 3.3.

For a G-algebra (A,α) we define its unitization to be the G-algebra (A+,α+):=(A⊕C*(E),α⊕τ) as described in Lemma 3.2.

Because G has a unit, A⁺ is unital. Actually, this is the only reason and place where we need a unit in G.

The aim of this section is the proof of Proposition 1.1.

Lemma 4.1.

Let D be a G-algebra and

an exact sequence of G-algebras. If [φ,E,T]∈KKG(D,B) such that f*(φ,E,T) is a degenerate Kasparov cycle in KKG(D,C), then it is of the form

where [φ′,E′,T′]∈KKG(D,A) with E′={ξ∈E|⟨ξ,ξ⟩∈j(A)}⊆E, φ′=φ(⋅)| E ′ and T′=T| E ′ .

Proof. The canonical proof of [11, Lemma 3.2] works verbatim also G-equivariantly.

Definition 4.2.

We recall that F(−)=KKG(A,−) denotes the functor F:C*→Ab with F(B)=KKG(A,B) and F(f)=f*(z)=z⊗Bf*(1B) for f∈C*(B,C) and z∈KKG(A,B).

Lemma 4.3.

The functor F given by F(B)=KKG(A,B) from C* to the abelian groups is stable. That is, for any G-algebra (B⊗K,γ) and any minimal projection e∈K such that the associated corner embedding φ:B→B⊗K with φ(b)=b⊗e happens to be a G-equivariant *-homomorphism, the map F(φ) is invertible.

Proof. Notice that F(φ)=(⋅)⊗B[φ], where

where the G-action on B⊗eK is given by γ. We propose an inverse element for [φ] by

where m is the multiplication operator, the G-action on B⊗Ke is given by γ, and the B-valued inner product on B⊗Ke is defined by ⟨x,y⟩=φ−1(x*y). We are done when showing that [φ]⊗B⊗Kz=1B and z⊗B[φ]=1B⊗K in KK^G. We have

by the G-Hilbert B,B-module isomorphism determined by

for all bi∈B,ki∈K. Similarly, z⊗B[φ]=1B⊗K is computed by the G-Hilbert B⊗K,B⊗K-module isomorphism given by

Lemma 4.4.

The functor F given by F(B)=KKG(D,B) from C* to the abelian groups is split-exact. That is, given a split exact sequence

its image under F is canonically split-exact.

Proof. Let π:B→M(A)=LA(A) be the standard *-homomorphism π(b)(a)=j−1(bj(a)) associated to the exact sequence and notice that it is G-equivariant. Consider the G-Hilbert A-module A⊕A with grading ϵ(x,y)=(x,−y). We have an element {j}−1:=[φ,A⊕A,F]∈KKG(B,A), where F is the flip automorphism and φ:B→LA(A⊕A) is given by

Similarly, there is an element {s∘f}⊥:=[ψ,B⊕B,F′]∈KKG(B,B), where F' is the flip automorphism and ψ:B→LB(B⊗B) is determined by

It was checked in [7] that {s∘f}⊥=1B−(s∘f)*(1B). Lemma 4.1 shows that {s∘f}⊥=j*({j}−1).

Let D be another G-algebra, and x∈KKG(D,B) be in the kernel of f_*. Then

which is in the image of j_*. Thus the sequence

is split exact.

By Lemmas 4.3 and 4.4 and the evidence of homotopy invariance we obtain the main result of this section:

Corollary 4.5.

Proposition 1.1 is true.

When we shall later introduce a Cuntz picture of Kasparov theory, the corresponding transformation produces a G-action S on a G-Hilbert A-module A, which will be - synthetically - written as Sg=ug∘αg, where α denotes the C*-action on A. This ug=Sg∘αg−1 will be defined next:

Definition 5.1.

Let (A,α) be a G-algebra. An α-cocycle is a map u:G→M(A) such that the identities

hold in M(A) for all g and h in G.

Lemma 5.2.

Let u be an α-cocycle.

• (a) Then we have αgg−1=ug*ug=ugug*=ugg−1∈M(A). In particular, every u_g is a partial isometry and the source and range projection of u_g both agree with αgg−1 and are in the center of M(A).

• (b) In particular, every u_e = α_e is a self-adjoint projection in the center of M(A) for all e ∈ E.

• (c) We may replace the second identity of (5.1) by the identity αg¯(ug−1)=ug* without changing the definition of a cocycle.

Proof. Note that αgg−1 is a projection of the center of M(A). Hence, u_g is a partial isometry by the first identity of (5.1). Using only the identities (5.1), we have

which checks Lemma 5.2(c). The second identity of (5.1) is on the other hand easily obtained from this new identity. The identity ug*ug=ugug* follows now from the first identity of (5.1) and the identity of Lemma 5.2(c) through

Definition 5.3.

Given an α-cocycle u we write uαu* for the G-action (uαu*)g(a)=ugαg(a)ug* on A.

Definition 5.4.

For an α-cocycle u we introduce a G-action δu on M₂(A) under the formula

Notice that αe(a)ue*=αe(a)αe=αe(a) for every e ∈ E by Lemmas 5.2 and 2.6, such that δeu=idM2⊗αe. With that and Lemma 5.2 it is straighforward to check that δu is indeed a G-action.

In this section we shall see that the objects (A,α) and (A,uαu*) are isomorphic under a stable functor, where u denotes an α-cocycle.

Definition 6.1.

Consider the two corner embeddings SA(a)=a000 and TA(a)=000a which define G-equivariant *-homomorphisms SA:(A,α)→(M2(A),δu) and TA:(A,uαu*)→(M2(A),δu).

Definition 6.2.

Let F be a stable functor from C* to the abelian groups. Then define an isomorphism

That is, under a stable functor 'the actions α and u α u^* are isomorphic' and we can switch between them via u_# as we like.

Lemma 6.3.

Consider the stable functor F from C^* to the abelian groups defined by F(A)=KKG(D,A). Then the map u_# from the last definition and its inverse map u#−1 can be realized by multiplication with the following Kasparov elements:

respectively, where the occurring Hilbert A-modules are trivially graded and (αu)g(a)=αg(a)ug and (αu*)g(a)=αg(a)ug* denote their G-actions, respectively.

Proof. Since u#−1=(SA)*−1∘(TA)* the claim is that z−1=[TA]⊗M2(A)[SA]−1. The KK^G-inverse [SA]−1 may be represented by

where m denotes the multiplication operator. On the other hand [TA]=[TA,0001M2(A),0]∈KKG(A,M2(A)). Here, the Hilbert modules have trivial grading and the G-actions are given by restriction of δ^u. We have an isomorphism

of G-Hilbert A,A-bimodules, where the image is also equipped with the restricted δ^u-action. This proves the claim. The case u_# is proven similarly.

Lemma 6.4.

Let φ:(A,α)→(B,β) be an equivariant *-homomorphism. Let u be an α-cocycle and v a β-cocycle such that vgφ(a)=φ(uga) for all g ∈ G and a ∈ A. Let F be a stable functor from C^* to Ab. Then φ is also an equivariant *-homomorphism φ:(A,uαu*)→(B,vβv*) such that

Proof. Just note that idM2⊗φ:(M2(A),δu)→(M2(B),δv) is an equivariant *-homomorphism satisfying (idM2⊗φ)∘SA=SB∘φ and (idM2⊗φ)∘TA=TB∘φ.

Until Section 10 assume that B is stable (i.e. B≅B⊗K)!

The A,B-cocycles defined next will serve as a Cuntz-picture of Kasparov cycles. We shall prove in the next section that they may substitute Kasparov theory. Confer Remark 8.4 for a motivation of the following definition:

Definition 7.1.

Let (A,α) and (B,β) be G-algebras (where B is stable). An A,B-cocycle is a quadruple

where

• (a) u₊ and u_- denote β-cocycles G→M(B),

• (b) φ± denote G-equivariant *-homomorphisms A→(M(B),u±β¯u±*), respectively,

• (c) φ+(a)−φ−(a)∈B,

• (d) u+g−u−g∈B

for all a in A and g in G.

Definition 7.2.

Two A,B-cocycles (φ±,u±) and (ψ±,v±) are isomorphic when there exists a G-equivariant automorphism γ∈Aut(B,β) such that

In the rest of the paper we shall identify isomorphic A,B-cocycles.

Definition 7.3.

The set of isomorphism classes of A,B-cocycles is denoted by EG(A,B).

Definition 7.4.

An A,B-cocycles (φ±,u±) is called degenerate if φ±=0. The set of degenerate A,B-cocycles is denoted by DG(A,B).

Definition 7.5.

Two A,B-cocycles (φ±t,u±t) (t=0,1) are called homotopic if there exists an A,B[0,1]-cocycle (φ±,u±) such that

where πt:B[0,1]→B denotes evaluation at time t=0,1.

Definition 7.6.

For (φ±,u±),(ψ±,v±)∈EG(A,B) define their sum to be

where V1,V2∈M(B) are G-invariant isometries such that V1V1*+V2V2*=1 (see Lemma 2.13).

Lemma 7.7.

Up to homotopy of A,B-cocycles, the last definition of sum of A,B-cocycles does not depend on the choice of the isometries V₁,V₂.

Proof. Let W1,W2∈M(B) be another pair of G-invariant isometries such that W1W1*+W2W2*=1. Then U=W1V1*+W2V2* defines a G-invariant unitary in M(B) such that UVi=Wi (i=0,1). By Lemma 2.14, U may be connected to 1 by a G-invariant, strictly continuous path (Ut)t in M(B). Then the cocycle

in EG(A,B[0,1]) yields the desired homotopy.

Definition 7.8.

Let FG(A,B) denote the quotient EG(A,B)/~ under the equivalence relation ~ defined by x1~x2 for x1,x2∈EG(A,B) if and only if there exists degenerate d1,d2∈DG(A,B) such that x1+d1 is homotopic to x2+d2. We equip FG(A,B) with the addition [x1]+[x2]:=[x1+x2].

In this section we shall isomorphically substitute Kasparov theory by its Cuntz picture in form of A,B-cocycles. This transition is given as follows:

Definition 8.1.

There is a map

where the G-Hilbert B-module B⊕B is equipped with the grading ϵ(x,y)=(x,−y) and the G-action

F is the flip automorphism, and the operator φ:A→LB(B⊕B) is defined by

Lemma 8.2.

The just defined Δ(φ±,u±) is indeed a Kasparov group element.

Proof. Denoting u=(u+,u−), γ=(β,β) and the indicated G-action (u+β,u−β) on B⊕B by W=uγ, one has

because of the third identity of (5.1) and because βg−1g is in the center of M(B).

We have