New York Journal of Mathematics

New York J. Math. 15 (2009) 505–531.

Equivariant KK-theory for semimultiplicative sets

Bernhard Burgstaller

Abstract. A semimultiplicative set G is a set which has a partially defined associative multiplication. We associate a reduced C^{∗}-algebra C_{r}^{∗}(G) to G and define reduced crossed products A ⋊ G. Moreover, we introduce a G-equivariant KK-theory and show the existence of a Kasparov product.

Contents

1. Introduction 505 2. Semimultiplicative sets 507 3. Crossed products 509 4. Equivariant KK-theory 511 5. Kasparov’s technical theorem 516 6. The Kasparov cap product 520 7. The Kasparov cup-cap product 526 References 530

1. Introduction

In this work we introduce and analyze some rudiments of semimultiplicative sets in connection with C^{∗}-algebras. Semimultiplicative sets appear in [1], and the somewhat stronger notion of a semigroupoid is due to Exel [3]. A semimultiplicative set G is a set which is endowed with a partially defined associative multiplication (Definition 1). That means we allow, as in groupoids, that a product xy may or may not be defined. When G is a group then there exists a left regular representation λ : G → B(ℓ^{2}(G)). In

505^{ISSN 1076-9803/09}

Received January 20, 2009; revised November 4, 2009.

Mathematics Subject Classification. 19K35, 20N02.

Key words and phrases. Equivariant KK-theory, semimultiplicative set, semigroup.

The author was supported by the Marie Curie Network MRTN-CT-2006-031962.

a similar way we define a left regular representations for semimultiplicative sets G (Definition 2), thereby canceling all invalid multiplications in G. This concept can also be used to define a reduced product A ⋊_{α} G for any left G-C^{∗}-algebra A, that is, a C^{∗}-algebra A which is endowed with a morphism α : G → End(A). (Technically, one uses the right regular representation in this case.)

In the second and final part of this work we focus on the K-theory of G-C^{∗}-algebras. We introduce a G-equivariant KK-theory KK^{G}(A, B) (Definitions 16, 17 and 22) for Hilbert C^{∗}-algebras A, B and discrete countable semimultiplicative sets G. A Hilbert C^{∗}-algebra A is a C^{∗}-algebra which is endowed with a left action G → End(A) and a right action G → End(A) under which A becomes a G-Hilbert A-module in the natural way. When G happens to be a group, then any G-C^{∗}-algebra is a Hilbert C^{∗}-algebra and our equivariant KK-theory has a similarity to Kasparov’s equivariant KK-theory for discrete groups G, the difference being that the underlying G-actions on Hilbert modules need not be full but degenerate (a “unit problem” so to say). See Lemma 7 and its preceding paragraph for the details. Our main work is to prove the existence of a Kasparov product for KK^{G} (Theorems 2 and 3), and to show its functoriality and associativity (Section 7).

An ongoing study of continuous semimultiplicative sets and their crossed products seems to be necessary to find the right continuity assumptions in KK^{G}, and actually we aim to continue our investigation in this direction. Kasparov’s equivariant KK-theory [7] was generalized by Le Gall for groupoids G in [11] and [12], see also Tu [15] for an overview. Since discrete semimultiplicative sets generalize discrete groupoids it is tempting to compare Le Gall’s theory with ours when G is a groupoid (though there seems to be an obvious difference already in the group case), but we will not go into that in this paper.

We give a brief overview of this paper. The Sections 2 and 3 are dedicated to semimultiplicative sets, some of their basic examples, and their crossed products. In Sections 4–6 we introduce { \atop KK^{G}}(A, B) (cycles divided out by operator equivalence) for left G-C^{∗}-algebras A, B and prove the existence of

Equivariant KK-theory for semimultiplicative sets 507

a Kasparov product for { \atop KK^{G}}. By definition we require a left and a right Gaction for Hilbert modules, though, we are only provided with a left action for the C^{∗}-algebras. This anomaly turns out to be a weak point in the theory, with bad functorial properties, whence in the last Section 7 we consider KK^{G}(A, B) exclusively for Hilbert C^{∗}-algebras A and B. Comparing the category of G-C^{∗}-algebras and Hilbert C^{∗}-algebras, the latter one seems be the “smooth” one when working in KK^{G} (at least in the approaches presented here). In the KK-theory part of this paper we closely follow Kasparov’s exposition in [8] and Skandalis’ paper [14]. This sometimes goes without saying. Influencing in general was also Valette’s book [16], which is also recommendable as an introduction to KK-theory.

2. Semimultiplicative sets

Definition 1. A semimultiplicative set G is a set which is endowed with a partially defined associative multiplication, that is, there exists a subset G^{(2)} ⊆ G×G and a multiplication G^{(2)} → G, (a, b) ↦ ab such that whenever (ab)c or a(bc) is defined then both (ab)c and a(bc) are defined and are equal.

If one also requires in the last definition that (ab)c is defined whenever both ab and bc are defined then one would speak of a semigroupoid, see Exel [3]. Let L_{g} denote the left multiplication operator on G, that is, L_{g}(h) = gh for g, h ∈ G. Its domain is {h | gh is defined}. Write R_{g} for the right multiplication operator. We say that G has injective left (resp. right) multiplication if L_{g} (resp. R_{g}) is injective for all g ∈ G. We write (e_{g})_{g∈G} for the canonical base in ℓ^{2}(G).

Definition 2. Assume that G has injective left multiplication. The left regular representation of G is the map λ : G → B(ℓ^{2}(G)) given by

λ_{g}(∑_{h∈G} α_{h}e_{h}) = ∑_{h∈G, gh is defined} α_{h}e_{gh},

where α_{h} are scalars in ℂ. The C^{∗}-subalgebra of B(ℓ^{2}(G)) generated by λ(G) is called the reduced C^{∗}-algebra of G and denoted by C_{r}^{∗}(G).

Analogously, for G with injective right multiplication we can define a right regular representation ρ : G → B(ℓ^{2}(G)) in the obvious way.

We are going to give some simple examples of semimultiplicative sets. Clearly, groups, groupoids, semigroups, semigroupoids and multiplicative sets are semimultiplicative sets. If R is a ring then R\{0} is a semimultiplicative set under multiplication (however not a semigroupoid in general). The set of natural numbers under addition is a semimultiplicative set (and semigroup), and its reduced C^{∗}-algebra is the Toeplitz algebra.

If Λ is a higher rank graph [9], that is one has a degree map d mapping Λ in ℕ^{k}, then the truncated graph Λ^{(≤N)} = {a ∈ S | d(a) ≤ N} is a semimultiplicative set (that is, a product ab is defined if and only if d(ab) ≤ N) which has injective left multiplication. This is not a semigroupoid.

For N ≥ 0, the real interval [0, N] is a semimultiplicative set (but not a semigroupoid) under addition. That means, we let the composition a ∘ b be defined if and only if a + b ∈ [0, N], and in this case we put a ∘ b = a + b. More generally, the interval [0, N · 1] in a C^{∗}-algebra is a semimultiplicative set.

Take the nonnegative reals ℝ_{+} and the compact interval [0, N] of reals whose elements are formally written as tμ for t ∈ [0, N]. Then ℝ_{+} ⊔ [0, N] is a semimultiplicative set (but not a semigroupoid) under the composition a∘b = a+b if both a and b are in ℝ_{+}, and a∘bμ = (a+b)μ if a ∈ ℝ_{+}, b ∈ [0, N] and a + b ≤ N. Other compositions are not allowed.

If G is a semimultiplicative set with injective multiplication, and λ is the left regular representation, then the set of nonzero words in the letters λ(G) ∪ λ(G)^{∗} is a semimultiplicative set G^{(∗)}. Though one is now provided with a left inverse λ(s)^{∗} for λ(s), in general G^{(∗)} need not to be a groupoid, see the next examples.

Consider the two graphs

\xymatrix{•a \ar[d]_{g} & •b \ar[d]_{g} &  & •a \ar[dr]^{h} \\ •c & •d \ar[u]_{h} & •b \ar[r] & •c \ar[r]_{h} & •d}

Assume the left graph is realized in a C^{∗}-algebra where a, b, c, d are mutually orthogonal projections, and g, h are partial isometries with source and range projections as indicated in the diagram (s(g) = a + b, r(g) = c + d). Let G be the semimultiplicative set which consists of all nonzero products in the letters {a, b, s, r, g, h}. Then G is not a semigroupoid since ga ≠ 0 and hg ≠ 0 but hga = 0. The semimultiplicative set G associated to the right graph has injective left multiplication and is a semigroupoid but not a groupoid, as naturally choosing the source and range maps causes a problem: the composition ha exists but s(h) = a + c ≠ a = r(a).

Take a semimultiplicative set G and a family (G_{i})_{i∈I} of copies of G. Set H = ⨆_{i∈I} G_{i} and write π_{i} : G → G_{i} for the canonical bijection. We define a multiplication on H by π_{i}(x)π_{j}(y) = π_{j}(xy) for all x, y ∈ G, i, j ∈ I and whenever xy is defined. Then H is a semimultiplicative set which has injective left multiplication if G does so. The left regular representation λ, however, is not injective, as L_{π_{i}(x)} = L_{π_{j}(x)} for all x ∈ G, i, j ∈ I.

Though the emphasis of this paper lies on discrete semimultiplicative sets, we will give one continuous example. Take the real interval [0, N] as our semimultiplicative set G as described above. The formal convolution

a ∗ b = “ ∫_{[0,N]} a(s)δ_{s}ds ∫_{[0,N]} b(t)δ_{t}dt”

leads us to the convolution product

(a ∗ b)(t) = ∫_{0}^{t} a(s)b(t − s)ds.

It is straightforward to check that this convolution product is associative. So the continuous left regular representation λ : C(G) → B(L^{2}[0, N]) given by λ(f) = f ∗ g for g ∈ L^{2}[0, N] yields a reduced C^{∗}-algebra C_{r}^{∗}(G) = C^{∗}(λ(C(G))) associated to the continuous semimultiplicative set G.

3. Crossed products

For the rest of this paper G denotes a semimultiplicative set, which is occasionally regarded as a discrete topological set.

Definition 3. A morphism (resp. antimorphism) σ : G → H between two semimultiplicative sets G and H is a map satisfying σ(gh) = σ(g)σ(h) (resp. σ(gh) = σ(h)σ(g)) for all (g, h) ∈ G^{(2)}.

Definition 4. If X is a linear space then a left (resp. right) linear action α on X is a morphism (resp. antimorphism) α : G → L(X), where L(X) denotes the set of linear maps on X. If X is a C^{∗}-algebra then an action α on X is a linear action on X such that α(g) is a ∗-homomorphism for all g ∈ G. In this case we call X a left (resp. right) G-C^{∗}-algebra.

Definition 5. A left action of a semimultiplicative set G on a set X is given by a subset (G×X)^{(2)} ⊆ G×X and a multiplication (G×X)^{(2)} → X, (g, x) ↦ gx such that (gh)x is defined if and only if g(hx) and gh is defined, and then (gh)x = g(hx), for all g, h ∈ G, x ∈ X.

Example 1. (a) Suppose G has injective left multiplication, and X is a discrete set endowed with a left action by G. We obtain a right G-action on the C^{∗}-algebra C_{0}(X) by letting

(fg)(x) = 1_{{gx is defined}}f(gx)

for f ∈ C_{0}(X) and g, x ∈ G. Similarly, a left action on C_{0}(X) is given by

(gf)(gx) = f(x), (gf)(y) = 0 if y ≠ gx.

(b) Analogously, if G has injective right multiplication and X has a right G-action then C_{0}(X) is a left and right G-C^{∗}-algebra by

(gf)(x) = 1_{{xg is defined}}f(xg),
(fg)(xg) = f(x), (fg)(y) = 0 if y ≠ xg.

Definition 6. Assume that G has injective left multiplication. Suppose that A is a C^{∗}-algebra which is endowed with a right G-action and which is essentially represented on a Hilbert space H. Let U : G → B(ℓ^{2}(G, H)) and the C^{∗}-representation π : A → B(ℓ^{2}(G, H)) be given by

π(a)(ξe_{h}) = ((ah)ξ)e_{h}, U_{g}(ξe_{h}) = 1_{{gh is defined}}ξe_{gh},

where ξe_{g} stands for the function h ↦ 1_{{h=g}}ξ (g, h ∈ G, ξ ∈ H, a ∈ A). Then the reduced crossed product G ⋉_{r} A is defined as the C^{∗}-subalgebra of B(ℓ^{2}(G, H)) generated by U_{G}π(A) = {U_{g}π(a) | a ∈ A, g ∈ G}.

Definition 7. Assume that G has injective right multiplication. Suppose that A is a left G-C^{∗}-algebra essentially represented on H. Let V, U : G → B(ℓ^{2}(G, H)) and π : A → B(ℓ^{2}(G, H)) be given by

π(a)(ξe_{h}) = ((ha)ξ)e_{h}, V_{g}(ξe_{h}) = 1_{{hg is defined}}ξe_{hg}, U_{g} = V_{g}^{∗},

(g, h ∈ G, ξ ∈ H, a ∈ A). Then the reduced crossed product A⋊_{r} G is defined as the C^{∗}-subalgebra of B(ℓ^{2}(G, H)) generated by

π(A)U_{G} = {π(a)U_{g} | a ∈ A, g ∈ G}.

If A = ℂ and the action of G on ℂ is trivial, that is, ag = a for all a ∈ A, g ∈ G, then the reduced C^{∗}-algebra C_{r}^{∗}(G) coincides with the reduced crossed product G⋉_{r}ℂ. If π′ : A → B(H′) is another essential representation then a canonical unitary W : ℓ^{2}(G, H) → ℓ^{2}(G, H′) shows that the definition of the reduced product does not depend on the representation π up to ∗- isomorphism.

Definition 8. A left action of G on a Hilbert space H is a morphism U : G → B(H) such that each U_{g} is a partial isometry (g ∈ G). We call H with such an action U a (left) G-Hilbert space. The action is called strong if U_{g}U_{h} = 0 for all undefined compositions gh.

Definition 9. If A is a right G-C^{∗}-algebra and H a left G-Hilbert space, then a ∗-homomorphism π : A → B(H) is called equivariant if

U_{g}^{∗}U_{g}π(ag) = U_{g}^{∗}π(a)U_{g}, π(a)U_{g}U_{g}^{∗} = U_{g}U_{g}^{∗}π(a), π(a)U_{g}^{∗}U_{g} = U_{g}^{∗}U_{g}π(a)

for all a ∈ A, g ∈ G. If the action on A is from the left then the first identity has to be replaced by U_{g}U_{g}^{∗}π(ga) = U_{g}π(a)U_{g}^{∗}.

It is easily verified that π of Definition 6 is an equivariant representation, and the action U on ℓ^{2}(G, H) is strong. If the action on A satisfies (ag)h = 0 for all a ∈ A whenever gh is not defined (g, h ∈ G) then one would have U_{g}^{∗}π(a)U_{g} = π(ag) for all a ∈ A, g ∈ G for the representation of Definition 6. However, this requirement is too restrictive for us as we also want to consider the trivial action on ℂ.

The next lemma links convolution algebras and equivariant representations.

Lemma 1. Endowing C_{c}(G, A) with the convolution product given by

(g · a)(h · b) = gh · ((ah)b)1_{{gh is defined}},

where a, b ∈ A, g, h ∈ G, and g · a denotes the map h ↦ 1_{{h=g}}a, the map σ(g · a) = U_{g}π(a) extends to an algebra homomorphism from C_{c}(G, A) to span(U_{G}π(A)) for any equivariant representation (π, U) with strong action U.

Proof. Straightforward. □

Definition 10. A right (resp. left) G-action α : G → End(A) on a C^{∗}- algebra A is called left-invertible (resp. right-invertible) if for all g ∈ G there is a T_{g} ∈ End(A) such that α(h)α(g)T_{g} = α(h) for all h ∈ G for which gh (resp. hg) exists.

If G has injective right multiplication then we may introduce a virtual inverse g^{−1} for each g ∈ G, and write xg^{−1} = h if x = hg, and let xg^{−1} otherwise undefined. (In general one does not obtain a semimultiplicative set in this way; take for example [0, N] as a counterexample: [−N, N] is not a semimultiplicative set.) If we suggestively write T_{g} = α(g^{−1}) in the last definition then it becomes clear that invertibility of a left G-action α is the counterpart to injective right multiplication in G.

Lemma 2. Assume that G has injective left (resp. right) multiplication and A is a right (resp. left) G-C^{∗}-algebra whose G-action is left-invertible (resp. right-invertible). (We will write g^{−1} := T_{g} for any choice T_{g} as in Definition 10.) Then the representation of Definition 6 (resp. 7) satisfies

U_{g}π(a)U_{g}^{∗} = π(g^{−1}(a))U_{g}U_{g}^{∗}

(resp. U_{g}^{∗}π(a)U_{g} = π(g^{−1}(a))U_{g}^{∗}U_{g}) for all a ∈ A, g ∈ G.

Proof. Straightforward. □

4. Equivariant KK-theory

In the rest of this paper all C^{∗}-algebras are supposed to be graded. A C^{∗}-algebra B is graded if there is a grading automorphism ε : B → B, ε^{2} = 1. The grading is called trivial if ε = 1. An element a ∈ B has degree i = 0, 1 if ε(b) = (−1)^{i}b. (Notation: ∂b = i.) All homomorphisms in the category of graded C^{∗}-algebras are graded, i.e., commute with ε. All commutators are graded, that is, [a, b] = ab − (−1)^{∂a·∂b}ab for homogenous elements a, b, and the commutator is extended by linearity to all a, b. A Hilbert module E over a C^{∗}-algebra B is always supposed to be graded, that is, there is a grading linear map ε : E → E, ε^{2} = 1, which is compatible with the grading of B, i.e., ε(xb) = ε(x)ε(b) and ε(⟨x, y⟩) = ⟨ε(x), ε(y)⟩, for all x, y ∈ E, b ∈ B. The space of linear maps L(E) on E is graded by the grading operator ε(T) = εT ε, T ∈ L(E). We write L(E) for the C^{∗}-algebra of adjointable operators T : E → E, and K(E) ⊆ L(E) for the C^{∗}-algebra of compact operators, that is, K(E) is generated by the elements θ_{ξ,η} ∈ L(E), θ_{ξ,η}(x) = ξ⟨η, x⟩, for all ξ, η ∈ E, see Kasparov [6] or the books [10], [5]. We write M(A) for the multiplier algebra of a C^{∗}-algebra A, see also [6], Theorem 1, for an isomorphism M(K(E)) ≅ L(E).

For the rest of this paper we (may) drop the associativity requirement on G, that is, G is only a set together with a subset G^{(2)} ⊆ G × G and a function G^{(2)} → G. However, we still call G a semimultiplicative set. All semimultiplicative sets G are supposed to be discrete and countable (thus locally compact, σ-compact Hausdorff spaces). All algebras and C^{∗}- algebras are left G-C^{∗}-algebras (if nothing else is said). All homomorphisms σ : A → B between C^{∗}-algebras A, B are supposed to be ∗-homomorphisms which are graded (i.e., commute with ε) and equivariant (i.e., σ(ga) = gσ(a) for all g ∈ G, a ∈ A).

Definition 11. Let B be a G-C^{∗}-algebra. An action of G on a Hilbert B-module E consists of a left linear G-action U : G → L(E) on E, and a right linear G-action V : G → L(E), which we denote by U^{∗} = V, satisfying

U_{g}U_{g}^{∗}U_{g} = U_{g}, U_{g}^{∗}U_{g}U_{g}^{∗} = U_{g}^{∗},
⟨U_{g}x, y⟩ = g⟨x, U_{g}^{∗}y⟩, U_{g}(xb) = (U_{g}x)(gb)

for all x, y ∈ E, b ∈ B, g ∈ G. U_{g} and U_{g}^{∗} must respect the grading (i.e., commute with ε) for each g ∈ G. Further we require g to be isometric on

B_{g} = \overline{span}\{⟨U_{g}^{∗}U_{g}x, y⟩ ∈ B | x, y ∈ E\}

for all g ∈ G. Given such maps U and V we call E a G-Hilbert B-module.

One may observe that B_{g} is a two-sided closed ideal (without G-action) in B, see Lemma 3 below. Notice that Definition 11 consistently redefines G-Hilbert spaces when B = ℂ with the trivial action and grading.

Lemma 3. Let E be a G-Hilbert module with action U. Then each U_{g} is a partial isometry on E with self-adjoint source and range projections U_{g}^{∗}U_{g} and U_{g}U_{g}^{∗} respectively in L(E), and inverse partial isometry U_{g}^{∗}. Moreover, ⟨x, U_{g}y⟩ = g⟨U_{g}^{∗}x, y⟩ and U_{g}^{∗}(xg(b)) = U_{g}^{∗}(x)b for all x, y ∈ E, g ∈ G, b ∈ B.

Proof. Let us begin with proving the following claim:

⟨x, U_{g}y⟩ = ⟨U_{g}y, x⟩^{∗} = (g⟨y, U_{g}^{∗}x⟩)^{∗} = g⟨U_{g}^{∗}x, y⟩.

Then one has g⟨U_{g}^{∗}U_{g}U_{g}^{∗}U_{g}x, y⟩ = g⟨U_{g}^{∗}U_{g}x, U_{g}^{∗}U_{g}y⟩, and by injectivity of g on B_{g} this shows that ⟨U_{g}^{∗}U_{g}x, y⟩ = ⟨U_{g}^{∗}U_{g}x, U_{g}^{∗}U_{g}y⟩. This shows that U_{g}^{∗}U_{g} is selfadjoint and hence in L(E). Each U_{g} is a partial isometry, that means, ∥U_{g}(U_{g}^{∗}U_{g}x)∥ = ∥U_{g}^{∗}U_{g}x∥ and U_{g}(1 − U_{g}^{∗}U_{g}) = 0. The last claim follows from U_{g}^{∗}U_{g}U_{g}^{∗}(xg(b)) = U_{g}^{∗}(U_{g}U_{g}^{∗}(x)g(b)) = U_{g}^{∗}(U_{g}(U_{g}^{∗}(x)b)). □

Definition 12. A Hilbert C^{∗}-algebra A is a G-C^{∗}-algebra which is also a G-Hilbert module over A with inner product ⟨x, y⟩ = x^{∗}y and action U_{g}(x) = g(x) for all x ∈ A, g ∈ G. We also require that U_{g}^{∗} is a ∗-homomorphism for all g ∈ G.

The algebra C_{0}(X) of Example 1 is a Hilbert C^{∗}-algebra. Any C^{∗}-algebra A with trivial action g(a) = a, a ∈ A, g ∈ G, is a Hilbert C^{∗}-algebra.

Definition 13. Given a G-Hilbert module E, we endow L(E) with the left linear action g(T) = U_{g}T U_{g}^{∗} and the right linear action g^{−1}(T) = U_{g}^{∗}T U_{g} for g ∈ G, T ∈ L(E).

L(E) and subalgebras of it are usually not regarded as G-algebras, as the action is not a C^{∗}-action. Note that g^{−1}(T) is indeed adjointable: from g⟨U_{g}^{∗}T U_{g}x, y⟩ = g⟨x, U_{g}^{∗}T^{∗}U_{g}y⟩ for all x, y ∈ E, g ∈ G, the injectivity of g on B_{g} and self-adjointness of U_{g}^{∗}U_{g} it follows ⟨U_{g}^{∗}T U_{g}x, y⟩ = ⟨x, U_{g}^{∗}T^{∗}U_{g}y⟩. With Lemma 3 one checks that g(T), g^{−1}(T) ∈ K(E) for all g ∈ G and compact operators T ∈ K(E).

Definition 14. A subalgebra A of L(E) is called G-invariant if for all g ∈ G the sets g(A), g^{−1}(A), U_{g}U_{g}^{∗}A, AU_{g}U_{g}^{∗}, U_{g}^{∗}U_{g}A, AU_{g}^{∗}U_{g} are subsets of A.

Definition 15. If A is a left G-C^{∗}-algebra and E is a G-Hilbert B-module, then a ∗-homomorphism π : A → L(E) is called equivariant if

U_{g}U_{g}^{∗}π(ga) = U_{g}π(a)U_{g}^{∗},
U_{g}U_{g}^{∗}π(a) = π(a)U_{g}U_{g}^{∗}, U_{g}^{∗}U_{g}π(a) = π(a)U_{g}^{∗}U_{g}

for all a ∈ A, g ∈ G. Moreover, we require that U_{g}^{∗}π(A)U_{g} ⊆ U_{g}^{∗}U_{g}π(A) for all g ∈ G (‘G^{−1}-invariance’).

In the rest of this article all Hilbert modules are supposed to be G-Hilbert modules, and all homomorphisms from C^{∗}-algebras into L(E) are supposed to be equivariant. We call a Hilbert B-module E together with an equivariant ∗-homomorphism φ : A → L(E) a Hilbert (A, B)-bimodule. (Notice that a · x := φ(a)(x) makes E a left A-module.) With some abuse of notation we shall often identify elements of A with operators on E.

Example 2. If ℂ is endowed with the trivial action and E is a G-Hilbert B-module then E is a G-Hilbert (ℂ, B)-bimodule.

Any C^{∗}-algebra A with the trivial action is a Hilbert (A, A)-bimodule. Consider C^{∗}-algebras A, B (without G-action) and a homomorphism σ :

A → B. Let X and G be as in Example 1. Then A_{1} = C_{0}(X, A) ≅ C_{0}(X) ⊗ A and B_{1} = C_{0}(X, B) are Hilbert C^{∗}-algebras, and C_{0}(X, B) is a Hilbert (A_{1}, B_{1})-bimodule with A_{1}-action (ab)(x) = σ(a(x))b(x) (a ∈ A_{1}, b ∈ B_{1}, x ∈ X).

Somewhat more generally, one may consider a family B = (B_{x})_{x∈X} of C^{∗}-algebras with a family of isomorphisms ϕ_{gx,x} : B_{x} → B_{gx} whenever gx is defined (g ∈ G) such that ϕ_{hgx,gx} ∘ ϕ_{gx,x} = ϕ_{hgx,x} whenever (hg)x is defined. Then the (continuous) sections Γ_{0}(B) of B vanishing at infinity are a Hilbert C^{∗}-algebra under the G-action β_{g}(b_{x}δ_{x}) = 1_{{gx is defined}}ϕ_{gx,x}(b_{x})δ_{gx}. One may also consider another C^{∗}-family A = (A_{x}; ψ_{x})_{x∈X} and a family of homomorphisms σ_{x} : A_{x} → B_{x} (x ∈ X) satisfying ϕ_{gx,x}σ_{x} = σ_{gx}ψ_{gx,x} to obtain a G-Hilbert (Γ_{0}(A), Γ_{0}(B))-bimodule Γ_{0}(B).

If G has injective right multiplication and A is a C^{∗}-algebra with invertible left G-action then the representation (π, U) of Definition 7 is equivariant in the sense of Definition 15 by Lemma 2.

Let S be an inverse semigroup and α an S-action on a C^{∗}-algebra A in the sense of Sieben [13], i.e., a morphism of S into the partial actions on A. Assume there exist commuting Hilbert-module-self-adjoint projections Q_{ss^{∗}}, Q_{s^{∗}s} ∈ End(A) projecting onto the range and source, respectively, of α_{s} (s ∈ S). Then β_{s} = α_{s}Q_{s^{∗}s} ∈ End(A) is a S-Hilbert C^{∗}-action on A. Indeed, note that (α_{t} ∘ α_{s})Q_{s^{∗}t^{∗}ts} = α_{t}Q_{t^{∗}t}α_{s}Q_{s^{∗}t^{∗}ts} (s, t ∈ S), and so Q_{s^{∗}t^{∗}ts} = α_{s^{∗}}Q_{t^{∗}t}α_{s}Q_{s^{∗}t^{∗}ts}. Hence α_{t}Q_{t^{∗}t}α_{s}Q_{s^{∗}s} = α_{t}Q_{t^{∗}t}Q_{ss^{∗}}α_{s}Q_{s^{∗}s} = α_{t}α_{s}α_{s^{∗}}Q_{t^{∗}t}α_{s}Q_{s^{∗}s} = α_{ts}Q_{s^{∗}t^{∗}ts}.

For the definitions of the internal and skew tensor products of Hilbert modules see Kasparov [7], Section 2. The grading operator for tensor products of Hilbert modules or C^{∗}-algebras is the diagonal grading operator ε ⊗ ε. We denote the skew commutative (minimal) tensor product between C^{∗}-algebras A, B (see Kasparov [7], Section 2) by A ⊗ B (that is (a_{1} ⊗ b_{1})(a_{2} ⊗ b_{2}) = (−1)^{∂b_{1}·∂a_{2}}(a_{1}a_{2} ⊗ b_{1}b_{2}), (a ⊗ b)^{∗} = (−1)^{∂a·∂b}(a^{∗} ⊗ b^{∗}) for a, a_{i} ∈ A, b, b_{i} ∈ B). We endow A ⊗ B with the diagonal action g(a ⊗ b) = g(a) ⊗ g(b) for all g ∈ G, a ∈ A, b ∈ B.

Lemma 4. If E_{i} are G-Hilbert B_{i}-modules (i = 1, 2) and φ : B_{1} → L(E_{2}) is an equivariant ∗-homomorphism (not necessarily satisfying the G^{−1}-invari-ance) then the internal tensor product E_{1} ⊗_{B_{1}} E_{2} is a G-Hilbert B_{2}-module under the diagonal-action U^{(1)} ⊗U^{(2)}. If E_{1} is a G-Hilbert (A, B_{1})-bimodule, then E_{1} ⊗_{B_{1}} E_{2} is a G-Hilbert (A, B_{2})-bimodule (under the A-action π : A → L(E_{1} ⊗_{B_{1}} E_{2}), π(a) = a ⊗ 1).

Proof. Consider the algebraic tensor product E_{1} ⊙E_{2} with its natural structure of a B_{2}-module and with the B_{2}-scalar product given by the formula

⟨x_{1} ⊙ x_{2}, y_{1} ⊙ y_{2}⟩ = ⟨x_{2}, φ(⟨x_{1}, y_{1}⟩)y_{2}⟩

for all x_{1}, y_{1} ∈ E_{1}, x_{2}, y_{2} ∈ E_{2}. Factoring out the B_{2}-submodule

N = {z ∈ E_{1} ⊙ E_{2} | ⟨z, z⟩ = 0}

and then completing the factor module in the norm ∥z∥ = ∥⟨z, z⟩∥^{1/2} we obtain a Hilbert B_{2}-module which is denoted by E_{1} ⊗_{B_{1}} E_{2}. This tensor product will be endowed with a G-action that comes from the diagonal action (U_{g}^{(1)} ⊙ U_{g}^{(2)})(x_{1} ⊙ x_{2}) = U_{g}^{(1)}(x_{1}) ⊙ U_{g}^{(2)}(x_{2}) on E_{1} ⊙ E_{2}, where the “adjoint” operator to W_{g} = U_{g}^{(1)} ⊙ U_{g}^{(2)} is given by W_{g}^{∗} = (U_{g}^{(1)})^{∗} ⊙ (U_{g}^{(2)})^{∗}. Indeed, it is straightforward to compute that

⟨(U_{g}^{(1)} ⊙ U_{g}^{(2)})(x_{1} ⊙ x_{2}), y_{1} ⊙ y_{2}⟩ = g⟨x_{1} ⊙ x_{2}, (U_{g}^{(1)}^{∗} ⊙ U_{g}^{(2)}^{∗})(y_{1} ⊙ y_{2})⟩

for all x_{1}, y_{1} ∈ E_{1}, x_{2}, y_{2} ∈ E_{2}, g ∈ G. It is also straightforward to check that (U_{g}^{(1)} ⊙ U_{g}^{(2)})^{∗}(U_{g}^{(1)} ⊙ U_{g}^{(2)}) is self-adjoint, idempotent and seminormcontractive on E_{1} ⊙ E_{2} by a similar argument usually used to show that L(E_{1}) ⊗ 1 ⊆ L(E_{1} ⊗ E_{2}) (see for instance Lance [10], Section 4). Hence for x, y ∈ E_{1} ⊙ E_{2} one has

⟨W_{g}x, W_{g}y⟩ = g⟨x, W_{g}^{∗}W_{g}y⟩ = g⟨W_{g}^{∗}W_{g}x, W_{g}^{∗}W_{g}y⟩,

and since g is isometric on ⟨W_{g}^{∗}W_{g}x, y⟩ ∈ span_{a,b}⟨U_{g}^{(2)}^{∗}U_{g}^{(2)}a, b⟩, one gets ∥W_{g}x∥ = ∥W_{g}^{∗}W_{g}x∥ ≤ ∥x∥, and consequently also ∥W_{g}^{∗}x∥ = ∥W_{g}W_{g}^{∗}x∥ ≤ ∥x∥ by W_{g}^{∗}W_{g}W_{g}^{∗} = W_{g}^{∗}. Thus W_{g} and W_{g}^{∗} leave N invariant and their linear quotient maps extend by continuity to linear maps on E_{1} ⊗_{B_{1}} E_{2} denoted by U_{g}^{(1)} ⊗U_{g}^{(2)} and U_{g}^{(1)}^{∗} ⊗U_{g}^{(2)}^{∗} which make E_{1} ⊗_{B_{1}} E_{2} a G-Hilbert module. □

Lemma 5. If E_{i} are G-Hilbert B_{i}-modules (i = 1, 2) then the skew tensor product E_{1} ⊗ E_{2} is a G-Hilbert (B_{1} ⊗ B_{2})-module under the diagonal action of G. If E_{i} are G-Hilbert (A_{i}, B_{i})-bimodules (i = 1, 2), then E_{1} ⊗ E_{2} is a G-Hilbert (A_{1} ⊗ A_{2}, B_{1} ⊗ B_{2})-bimodule.

Proof. This may be proved similarly as Lemma 4. We only discuss the G^{−1}- invariance of Definition 15: Let U denote the diagonal action on E_{1} ⊗ E_{2} and π : A_{1} ⊗ A_{2} → L(E_{1} ⊗ E_{2}) the canonical homomorphism. It is clear that U_{g}^{∗}π(A_{1} ⊙ A_{2})U_{g} ⊆ U_{g}^{∗}U_{g}π(A_{1} ⊗ A_{2}) =: X for all g ∈ G. But then also U_{g}^{∗}π(A_{1} ⊗ A_{2})U_{g} ⊆ X, as X is the closed image of the ∗-homomorphism σ, where σ(x) = U_{g}^{∗}U_{g}π(x). □

For a Hilbert (A, B)-bimodule E and a subset C ⊆ L(E), we denote

Q_{C}(E) = {T ∈ L(E) | [T, c] ∈ K(E), ∀c ∈ C},
I_{C}(E) = {T ∈ L(E) | T c and cT in K(E), ∀c ∈ C}.

Definition 16. Let A and B be G-C^{∗}-algebras. A cycle over (A, B) is a pair (E, T), where E is a countably generated G-Hilbert (A, B)-bimodule, and T is an operator in Q_{A}(E) of degree 1 such that

T − T^{∗}, T^{2} − 1, φ_{1}(g) = U_{g}T U_{g}^{∗} − U_{g}U_{g}^{∗}T U_{g}U_{g}^{∗} = g(T) − gg^{−1}(T), φ_{2}(g) = U_{g}U_{g}^{∗}T − T U_{g}U_{g}^{∗}, φ_{3}(g) = U_{g}^{∗}U_{g}T − T U_{g}^{∗}U_{g}

belong to I_{A}(E) for all g ∈ G. We shall not distinguish between cycles (E_{1}, T_{1}) and (E_{2}, T_{2}) if there is an isometric, grading preserving isomorphism u : E_{1} → E_{2} of G-Hilbert (A, B)-bimodules with T_{2} = uT_{1}u^{−1}. The set of all cycles will be denoted by 𝔼^{G}(A, B). A cycle (E, T) will be called degenerate if the elements

[a, T], a(T − T^{∗}), a(T^{2} − 1), aφ_{1}(g), aφ_{2}(g), aφ_{3}(g)

are 0 for all a ∈ A, g ∈ G. The set of degenerate cycles is denoted by D^{G}(A, B).

Lemma 6. If (E, T) ∈ 𝔼^{G}(A, B) then U_{g}^{∗}T U_{g} −U_{g}^{∗}U_{g}T U_{g}^{∗}U_{g} ∈ I_{A}(E) for all a ∈ A.

Proof. By the rules of Definition 15 it is straightforward to check that ag^{−1}(T) − ag^{−1}g(T) = g^{−1}(g(a)gg^{−1}(T) − g(a)g(T)) ∈ K(E). □

We define an addition of cycles (E_{1}, T_{1}), (E_{2}, T_{2}) ∈ 𝔼^{G}(A, B) by taking the direct sum: (E_{1}, T_{1}) ⊕ (E_{2}, T_{2}) = (E_{1} ⊕ E_{2}, T_{1} ⊕ T_{2}).

Definition 17. Two cycles (E_{0}, T_{0}) and (E_{1}, T_{1}) over (A, B) are operatorially homotopic if E_{0} = E_{1} and there exists a norm continuous path t ↦ T_{t} ∈ L(E_{0}) (t ∈ [0, 1]) such that for each t ∈ [0, 1] the pair (E_{0}, T_{t}) is a cycle over (A, B). Two cycles (E_{0}, T_{0}) and (E_{1}, T_{1}) in 𝔼^{G}(A, B) are operatorially equivalent if there are degenerate cycles (F_{0}, S_{0}), (F_{1}, S_{1}) ∈ D^{G}(A, B) such that (E_{0}, T_{0})⊕(F_{0}, S_{0}) is operatorially homotopic to (E_{1}, T_{1})⊕(F_{1}, S_{1}). The set { \atop KK^{G}}(A, B) is defined as the quotient of 𝔼^{G}(A, B) by the equivalence relation given by operatorial equivalence.

Proposition 1. { \atop KK^{G}}(A, B) is an abelian group with addition given by direct sum.

Proof. One proves this along the lines of [7], Section 4, Theorem 1, or [14], Proposition 4. □

We remark that the G-action of a Hilbert module can be completely degenerate to zero, and cycles of such Hilbert modules in the sense of Definition 16 coincide with cycles in the sense of Kasparov [8] for the trivial group. One may circumvent this difference by restricting to unital semimultiplicative sets G (possibly by adjoining a unit) and requiring that the unit of G always acts as the identity on C^{∗}-algebras and Hilbert modules. Otherwise we have the following elementary observation.

Lemma 7. If G is a group and E is a G-Hilbert module then U_{g}U_{g}^{∗} = U_{e}U_{e}^{∗} and U_{g}^{∗}U_{g} = U_{e}^{∗}U_{e} for all g ∈ G, and U_{e}^{∗} is the adjoint of U_{e} ∈ L(E). If ker(U_{e}) = 0 then U_{g}^{∗}U_{g} = U_{g}U_{g}^{∗} = 1 and U_{g}^{∗} = U_{g^{−1}} for all g ∈ E (and thus E is a G-Hilbert module in the sense of [7]).

Proof. The first claim follows from U_{h}U_{h}^{∗} = U_{gg^{−1}h}U_{h}^{∗} = U_{g}U_{g}^{∗}U_{g}U_{g^{−1}h}U_{h}^{∗} = U_{g}U_{g}^{∗}U_{h}U_{h}^{∗} and similarly U_{g}U_{g}^{∗} = U_{g}U_{g}^{∗}U_{h}U_{h}^{∗} for all g, h ∈ G. Further, U_{e} = (U_{e}U_{e}^{∗})(U_{e}^{∗}U_{e}) ∈ L(E) and its adjoint is (U_{e}^{∗}U_{e})(U_{e}U_{e}^{∗}) = U_{e}^{∗}. For the last claim, P = U_{e}^{∗}U_{e} = U_{g}^{∗}U_{g} is a full selfadjoint projection and hence P = 1. Moreover, by Lemma 3 all U_{g} are bijective and consequently U_{g}^{∗} = U_{g}^{−1} = U_{g^{−1}}. □

5. Kasparov’s technical theorem

If nothing else is said, approximate units are supposed to be positive, increasing and all their elements having degree 0. If A is a subalgebra and Δ a subset of an algebra B then Δ derives A if [a, d] ∈ A for all a ∈ A, d ∈ Δ. (All commutators are graded.) In this section we prove a modification of the so-called Kasparov technical theorem, see Kasparov [7], Section 3. We follow closely Kasparov [8], Section 1.4, a simplification of Kasparov’s original proof due to Higson [4]. If X is a locally compact Hausdorff space and A a C^{∗}- algebra then we also write A(X) for the C^{∗}-algebra C_{0}(X, A).

Lemma 8. Let E be a G-Hilbert module with G-action U, A a G-invariant σ-unital subalgebra of L(E), Y a σ-compact locally compact Hausdorff space, and φ : Y → L(E) a function such that [φ(y), a] ∈ A for all a ∈ A, y ∈ Y, and y ↦ [φ(y), a] is a continuous function on Y (norm topology in L(E)) for all a ∈ A. Then there is a countable approximate unit (u_{i}) ⊆ A for A such that the limits

lim_{i→∞} ∥[u_{i}, φ(y)]∥,

lim_{i→∞} ∥U_{g}u_{i}U_{g}^{∗} − u_{i}U_{g}U_{g}^{∗}∥, lim_{i→∞} ∥u_{i}U_{g}U_{g}^{∗} − U_{g}U_{g}^{∗}u_{i}∥,

lim_{i→∞} ∥U_{g}^{∗}u_{i}U_{g} − u_{i}U_{g}^{∗}U_{g}∥, lim_{i→∞} ∥u_{i}U_{g}^{∗}U_{g} − U_{g}^{∗}U_{g}u_{i}∥,

are 0 for all y ∈ Y, g ∈ G. These limits are uniform on compact subsets of Y and G respectively.

Proof. Let X_{1} ⊆ X_{2} ⊆ ⋯ be an increasing sequence of open sets in G, with compact closures and ⋃_{n} X_{n} = G. Let also Y_{1} ⊆ Y_{2} ⊆ ⋯ be a similar sequence in Y and (v_{i}) ⊆ A a (positive increasing) countable approximate unit for A. Using induction, suppose that we have already constructed u_{1} ≤ u_{2} ≤ ⋯ ≤ u_{n} out of finite convex linear combinations of elements of v_{i}, and the following conditions are fulfilled:

∥u_{k}v_{j} − v_{j}∥ ≤ 1/k, ∥[u_{k}, φ(y)]∥ ≤ 1/k, ∥u_{k}U_{g}U_{g}^{∗} − U_{g}U_{g}^{∗}u_{k}∥ ≤ 1/k,

∥u_{k}U_{g}^{∗}U_{g} − U_{g}^{∗}U_{g}u_{k}∥ ≤ 1/k, ∥g(u_{k}) − gg^{−1}(u_{k})∥ ≤ 1/k

for all j ≤ k, y ∈ Y_{k}, g ∈ X_{k}, k ≤ n. To construct u_{n+1}, note that v_{i} ≥ u_{n} for all i ≥ m for some m ≥ 1. Let Λ be the convex hull of {v_{m}, v_{m+1}, … }. Denote by Z the disjoint union of {v_{1}, …, v_{n+1}}, Y_{n+1} and three copies X_{n+1}^{(1)}, X_{n+1}^{(2)}, X_{n+1}^{(3)} of X_{n+1}. For any v ∈ Λ let a_{v} ∈ A(Z) be the function defined by

a_{v}(v_{j}) = vv_{j} − v_{j}, a_{v}(y) = [v, φ(y)], a_{v}(g) = U_{g}U_{g}^{∗}v − vU_{g}U_{g}^{∗},
a_{v}(h) = U_{h}^{∗}U_{h}v − vU_{h}^{∗}U_{h}, a_{v}(l) = l(v) − ll^{−1}(v)

for all 1 ≤ j ≤ n + 1, y ∈ Y_{n+1}, g ∈ X_{n+1}^{(1)}, h ∈ X_{n+1}^{(2)}, l ∈ X_{n+1}^{(3)}. Suppose that there is no element u_{n+1} ∈ Λ with the required properties. Since the set of functions {a_{v} | v ∈ Λ} is convex, the separation theorem gives a bounded linear functional f on A(Z) with |f(a_{v})| ≥ 1 for all v ∈ Λ. This leads to a contradiction in the following way.

Write B = L(E), and denote by B(Z)′′ and A(Z)′′ the universal enveloping von Neumann algebras of B(Z) and A(Z), respectively, and identify A(Z)′′ as a subset of B(Z)′′. Regarding v_{i} as an element in B(Z) (constant function with value v_{i}), we have v_{i} ↑ p in the weak operator topology for some element p ∈ B(Z)′′. Since Z is compact, by a simple compactness argument we see that v_{i} is an approximate unit for A(Z), and so p is a unit for A(Z)′′. Write φ′ ∈ B(Z) for the function φ′|_{Y_{n+1}} = φ|_{Y_{n+1}} and

φ′|_{Z\Y_{n+1}} = 0. Since [p, φ′] ∈ A(Z)′′,

[p, φ′] = [p^{2}, φ′] = p[p, φ′] + [p, φ′]p = 2[p, φ′],

which implies that [p, φ′] = 0. Define ψ(z) = U_{z}U_{z}^{∗} for z ∈ X_{n+1}^{(1)} ⊆ G and ψ(z) = 0 for other z. By the G-invariance of A, [p, ψ] ∈ A(Z)′′, and by the same argument as before we thus obtain [p, ψ] = 0. For a ∈ A(Z), write σ(a), σ^{−1}(a) ∈ A(Z) for the functions

σ(a)(z) = z(a(z)) = U_{z}a(z)U_{z}^{∗},
σ^{−1}(a)(z) = z^{−1}(a(z)) = U_{z}^{∗}a(z)U_{z},

for z ∈ X_{n+1}^{(3)} ⊆ G, and σ(a)(z) = σ^{−1}(a)(z) = 0 for other z. As

∥σ(v_{i})σσ^{−1}(a) − σσ^{−1}(a)∥ ≤ ∥v_{i}σ^{−1}(a) − σ^{−1}(a)∥,
∥σσ^{−1}(v_{i})σσ^{−1}(a) − σσ^{−1}(a)∥ ≤ ∥v_{i}σ^{−1}(1)a − σ^{−1}(1)a∥

for all a ∈ A(Z), and since σ^{−1}(a), σ^{−1}(1)a ∈ A(Z), the sequences σ(v_{i}) = σσ^{−1}σ(v_{i}) and σσ^{−1}(v_{i}) are (not necessarily increasing and positive) approximate units for the C^{∗}-subalgebra A_{σ} = \overline{σσ^{−1}(A(Z))} ⊆ A(Z). Hence the weak operator topology limits α, β of σ(v_{i}) and σσ^{−1}(v_{i}) in B(Z)′′ (if the sequence σ(v_{i}) does not converge, we go over to a weak operator topology convergent subsequence σ(v_{k_{i}})) are units of A′′_{σ}, and so α = β.

The above calculations show that the weak operator topology limit of a_{v_{i}} vanishes in A(Z)′′. Hence lim_{i} f(a_{v_{i}}) = 0 (by a well-known linear topological identification of A(Z)′′ with the bidual space A(Z)^{∗∗}), which is a contradiction. Obviously, the constructed sequence u_{k} satisfies the claim. □

In the next theorem we regard M(J) as a subalgebra of L(E), see [10], Proposition 2.1.

Theorem 1. Let E be a Hilbert module, J a nondegenerate σ-unital Ginvariant subalgebra of L(E), A_{1} a σ-unital G-invariant subalgebra of M(J), and A_{2} a σ-unital subalgebra (without G-action) of M(J). Let Δ be a norm-separable subset of M(J) which derives A_{1}. Let Ω be a σ-compact locally compact Hausdorff space, and φ, ψ : Ω → M(J) be bounded functions. Assume that

A_{1}A_{2}, A_{1}φ(Ω), ψ(Ω)A_{1} ⊆ J,

and the functions

ω ↦ aφ(ω), ω ↦ φ(ω)a, ω ↦ aψ(ω), ω ↦ ψ(ω)a

are continuous on Ω, with respect to the norm topology in M(J), for all a ∈ A_{1} + J. Then there are positive elements M_{1}, M_{2} ∈ M(J) of degree 0 such that M_{1} + M_{2} = 1,

M_{i}a_{i}, [M_{i}, d], M_{2}φ(ω), ψ(ω)M_{2} ⊆ J,
g(M_{i}) − gg^{−1}(M_{i}), [g(1), M_{i}], [g^{−1}(1), M_{i}] ⊆ J

for all a_{i} ∈ A_{i}, d ∈ Δ, g ∈ G, ω ∈ Ω (i = 1, 2), and the functions

ω ↦ M_{2}φ(ω), ω ↦ ψ(ω)M_{2}

are norm continuous on Ω.

Proof. The proof is similar to the proof of the Theorem of Subsection 1.4 on page 151 of Kasparov’s paper [8], with some adaption we discuss now. In Kasparov’s paper G is a group, and somewhere in the proof of the theorem one chooses approximate units (u_{i}) ⊆ A_{1} for A_{1} and (v_{i}) ⊆ J for J according to the lemma on page 152 in Kasparov’s paper satisfying (among other things)

∥g(u_{n}) − u_{n}∥ ≤ 2^{−n}, ∀n, ∀g ∈ \overline{X}_{n}, and (3)

∥g(b_{n}) − b_{n}∥ ≤ 2^{−n}, ∀n, ∀g ∈ \overline{X}_{n}, (6)

where b_{n} is defined by b_{n} = (v_{n} − v_{n−1})^{1/2}. The sought element M_{2} ∈ L(E) is defined as the series ∑_{n≥1} b_{n}u_{n}b_{n} which converges in the strict topology. By the estimates (3) and (6) one gets the estimate

∥g(b_{n}u_{n}b_{n}) − b_{n}u_{n}b_{n}∥ ≤ 3 · 2^{−n}

for all n ≥ 1 (see the bottom of page 153 in Kasparov’s paper).

We modify Kasparov’s proof as follows. At first, Kasparov’s stated theorem deals only with one function φ. But it is quite obvious how to modify the proof that one can handle both functions φ and ψ. Next, Kasparov’s function φ has domain G. At the beginning of the proof he writes G as G = ⋃_{n∈ℕ} X_{n} with open subsets X_{n} ⊆ G with compact closures. We modify the proof in that we also choose a union Ω = ⋃_{n∈ℕ} Ω_{n} of open subsets Ω_{n} ⊆ Ω with compact closures, and substitute X_{n} by Ω_{n} everywhere there where X_{n} acts as a domain of φ or ψ. Instead of the subset W_{n} ⊆ J defined under point (4) in Kasparov’s proof, we take

W_{n} = \{k, u_{n}h_{2}, u_{n+1}h_{2}\}∪u_{n}φ(\overline{Ω}_{n})∪u_{n+1}φ(\overline{Ω}_{n+1})∪ψ(\overline{Ω}_{n})u_{n}∪ψ(\overline{Ω}_{n+1})u_{n+1}.

Next, we choose the mentioned approximate units (u_{i}) and (v_{i}) by Lemma 8 in such a way that we have the estimates

∥g(u_{n}) − gg^{−1}(u_{n})∥ + ∥U_{g}U_{g}^{∗}u_{n} − u_{n}U_{g}U_{g}^{∗}∥ + ∥U_{g}^{∗}U_{g}u_{n} − u_{n}U_{g}^{∗}U_{g}∥ (3)

≤ 2^{−n}, ∀n, ∀g ∈ X_{n}, and

∥g(c_{n}) − gg^{−1}(c_{n})∥ + ∥U_{g}U_{g}^{∗}c_{n} − c_{n}U_{g}U_{g}^{∗}∥ + ∥U_{g}^{∗}U_{g}c_{n} − c_{n}U_{g}^{∗}U_{g}∥ (6′)

≤ (1/100)2^{−n}/N_{n}^{2}, ∀n, ∀g ∈ X_{n},

rather than the estimates (3) and (6) in Kasparov’s paper. Thereby denote c_{n} = b_{n}^{2}, let ∑_{k=0}^{∞} α_{k}(x − 1)^{k} = x^{1/2} be the power series of x^{1/2} at 1, and choose N_{n} such that ∑_{k=N_{n}+1}^{∞} |α_{k}| ≤ (1/100)2^{−n} for all n ∈ ℕ. Note that ∥b_{n} − ∑_{k=0}^{N_{n}} α_{k}(c_{n} − 1)^{k}∥ ≤ (1/100)2^{−n} for all n ∈ ℕ. From (6′) we thus deduce

∥g(b_{n}) − gg^{−1}(b_{n})∥ + ∥U_{g}U_{g}^{∗}b_{n} − b_{n}U_{g}U_{g}^{∗}∥ + ∥U_{g}^{∗}U_{g}b_{n} − b_{n}U_{g}^{∗}U_{g}∥ (6)

≤ 2^{−n}, ∀n, ∀g ∈ X_{n}

(mainly by similar estimates we show next). This leads one to the following estimate.

∥U_{g}b_{n}u_{n}b_{n}U_{g}^{∗} − U_{g}b_{n}U_{g}^{∗}U_{g}u_{n}U_{g}^{∗}U_{g}b_{n}U_{g}^{∗}∥
≤ ∥U_{g}(U_{g}^{∗}U_{g}b_{n} − b_{n}U_{g}^{∗}U_{g})u_{n}b_{n}U_{g}^{∗}∥
+ ∥U_{g}b_{n}U_{g}^{∗}U_{g}(U_{g}^{∗}U_{g}u_{n} − u_{n}U_{g}^{∗}U_{g})b_{n}U_{g}^{∗}∥
+ ∥U_{g}b_{n}U_{g}^{∗}U_{g}u_{n}U_{g}^{∗}U_{g}(U_{g}^{∗}U_{g}b_{n} − b_{n}U_{g}^{∗}U_{g})U_{g}^{∗}∥ ≤ 3 · 2^{−n}.

for all g ∈ G, n ∈ ℕ. A similar estimate yields

∥gg^{−1}(b_{n}u_{n}b_{n}) − gg^{−1}(b_{n})gg^{−1}(u_{n})gg^{−1}(b_{n})∥ ≤ 3 · 2^{−n}.

Hence

∥g(b_{n}u_{n}b_{n}) − gg^{−1}(b_{n}u_{n}b_{n})∥
≤ 6 · 2^{−n} + ∥g(b_{n})g(u_{n})g(b_{n}) − gg^{−1}(b_{n})gg^{−1}(u_{n})gg^{−1}(b_{n})∥
≤ 6 · 2^{−n} + ∥(g(b_{n}) − gg^{−1}(b_{n}))g(u_{n})g(b_{n})∥
+ ∥gg^{−1}(b_{n})(g(u_{n}) − gg^{−1}(u_{n}))g(b_{n})∥
+ ∥gg^{−1}(b_{n})gg^{−1}(u_{n})(g(b_{n}) − gg^{−1}(b_{n}))∥ ≤ 9 · 2^{−n}.

Also,

∥U_{g}U_{g}^{∗}b_{n}u_{n}b_{n} − b_{n}u_{n}b_{n}U_{g}U_{g}^{∗}∥ ≤ 3 · 2^{−n},
∥U_{g}^{∗}U_{g}b_{n}u_{n}b_{n} − b_{n}u_{n}b_{n}U_{g}^{∗}U_{g}∥ ≤ 3 · 2^{−n}.

Other things of Kasparov’s proof need not to be changed. □

Corollary 1. Let M_{1} and M_{2} be the operators of Theorem 1, and assume that J = K(E). Then all claims of Theorem 1 (excepting M_{1} +M_{2} = 1) hold also for M_{i}^{1/2} rather than M_{i} (i = 1, 2).

Proof. As g(M_{i}^{n}) ≡ g(M_{i})^{n} and g(1)M_{i}^{n} ≡ (g(1)M_{i})^{n} modulo J, writing M_{i}^{1/2} as a power series shows that g(M_{i}^{1/2}) − g(M_{i})^{1/2} ∈ J, g(1)M_{i}^{1/2} − (g(1)M_{i})^{1/2} ∈ J and [d, M_{i}^{1/2}] ∈ J for all d ∈ Δ. Hence, modulo J we get

g(M_{i}^{1/2}) − gg^{−1}(M_{i}^{1/2}) ≡ g(M_{i})^{1/2} − gg^{−1}(M_{i})^{1/2}

= (g(M_{i}) − gg^{−1}(M_{i}))(g(M_{i})^{1/2} + gg^{−1}(M_{i})^{1/2})^{−1}

≡ 0.

Next, M_{i}^{1/2}a_{i} = M_{i}^{−1/2}M_{i}a_{i} ∈ J, and similarly we treat the remaining cases. □

6. The Kasparov cap product

Given a G-Hilbert module E, we define A(E) as the smallest G-invariant C^{∗}-subalgebra of L(E) generated by 1. Note that ∂U_{g} = ∂U_{g}^{∗} = 0 for all g ∈ G, whence all elements of A(E) are zero graded.

Lemma 9. If π : A → L(E) is an equivariant representation then π(A) commutes with A(E).

If (E, T) ∈ 𝔼^{G}(A, B) then a[x, T] ∈ K(E) for all a ∈ A, x ∈ A(E).

Proof. Let W be the smallest set in L(E) which is closed under taking products, is invariant under G and G^{−1}, and contains 1. Since W is selfadjoint, A(E) = \overline{span}W. That π(A) commutes with W can be proved by induction for expressions in W. Suppose that x ∈ W and π(a)x = xπ(a) for all a ∈ A. Fix a ∈ A, g ∈ G. Then U_{g}^{∗}π(a)U_{g} = U_{g}^{∗}U_{g}π(b) for some b ∈ A by Definition 15, and thus π(a)U_{g} = U_{g}π(b) and π(b)U_{g}^{∗} = U_{g}^{∗}π(a). Hence, π(a)U_{g}xU_{g}^{∗} = U_{g}xU_{g}^{∗}π(a). This proves the first claim of the lemma.

Let (E, T) ∈ 𝔼^{G}(A, B). Take x ∈ W and assume that a(xT − T x) ∈ K(E) for all a ∈ A. Fix a ∈ A, g ∈ G, and write b = U_{g}^{∗}aU_{g} = U_{g}^{∗}U_{g}b′ for some b′ ∈ A by Definition 15. Multiplying

a(U_{g}T U_{g}^{∗} − T U_{g}U_{g}^{∗}) = U_{g}U_{g}^{∗}U_{g}b′T U_{g}^{∗} − aT U_{g}U_{g}^{∗} ∈ K(E)

from the right with U_{g} (thereby noticing that b′T U_{g}^{∗}U_{g} ≡ b′U_{g}^{∗}U_{g}T modulo K(E)), one gets a(U_{g}T − T U_{g}) ∈ (K(E)U_{g} + U_{g}K(E)). In a similar way, by multiplying a(U_{g}^{∗}T U_{g} − T U_{g}^{∗}U_{g}) ∈ K(E) (see Lemma 6) from the right with U_{g}^{∗}, one obtains a(U_{g}^{∗}T −T U_{g}^{∗}) ∈ (K(E)U_{g}^{∗} +U_{g}^{∗}K(E)). With these formulas, the formulas aU_{g} = U_{g}b, U_{g}^{∗}a = bU_{g}^{∗}, ax = xa, bx = xb, b = U_{g}^{∗}U_{g}b′, and the invariance of K(E) under G and G^{−1}, it is straightforward to compute that a(g(x)T −T g(x)) ≡ g(b′(xT −T x)) ≡ 0 modulo K(E). A similar computation shows that a(g^{−1}(x)T − T g^{−1}(x)) ∈ K(E). □

Definition 18. Let E_{1} be a Hilbert B_{1}-module, E_{2} a Hilbert (B_{1}, B_{2})- bimodule, and E_{12} = E_{1} ⊗_{B_{1}} E_{2}. For any ξ ∈ E_{1}, define θ_{ξ} ∈ L(E_{2}, E_{12}) by θ_{ξ}(η) = ξ ⊗ η (with an adjoint given by θ_{ξ}^{∗}(ξ_{1} ⊗ η) = ⟨ξ, ξ_{1}⟩ · η). Let T_{2} ∈ L(E_{2}). An element T_{12} ∈ L(E_{12}) will be called a T_{2}-connection on E_{12} if for any ξ ∈ E_{1} both

θ_{ξ}T_{2} − (−1)^{∂ξ·∂T_{2}}T_{12}θ_{ξ},
θ_{ξ}T_{2}^{∗} − (−1)^{∂ξ·∂T_{2}}T_{12}^{∗}θ_{ξ}

are in K(E_{2}, E_{12}). (Adjoining the last element gives T_{2}θ_{ξ}^{∗} − (−1)^{∂ξ·∂T_{2}}θ_{ξ}^{∗}T_{12}.)

Note that the definition of T_{2}-connections does not involve a G-structure. Connections were introduced by Connes and Skandalis in [2]. By Lemma 9, 1 ⊗ T is a well-defined operator in L(E_{12}) for T ∈ A(E_{2}).

Lemma 10. With the notation from the previous definition, let T_{12} be a T_{2}-connection.

If T_{2} ∈ I_{B_{1}}(E_{2}) then T_{12} ∈ I_{K(E_{1})⊗1}(E_{12}).

If (E_{2}, T_{2}) ∈ 𝔼^{G}(B_{1}, B_{2}) then

T_{12} ∈ Q_{K(E_{1})⊗A(E_{2})}(E_{12}), (1)

g(T_{12}) − gg^{−1}(T_{12}) ∈ I_{K(E_{1})⊗1}(E_{12}), (2)

(K(E_{1}) ⊗ 1)(g(1)T_{12}^{μ} − gg^{−1}(T_{12}^{μ})) ∈ K(E_{12}), (3)

(K(E_{1}) ⊗ 1)(g^{−1}(1)T_{12}^{μ} − g^{−1}g(T_{12}^{μ})) ∈ K(E_{12}) (4)

for all μ ∈ {1, ∗}.

If T_{1} ∈ L(E_{1}) then (K(E_{1}) ⊗ 1)[T_{12}, T_{1} ⊗ 1] ∈ K(E_{12}).

Proof. First notice that θ_{ξ,η} ⊗ 1 = θ_{ξ}θ_{η}^{∗} for all ξ, η ∈ E_{1}. Take ξ ∈ E_{1} and write b_{i} = ⟨ξ, ξ⟩(⟨ξ, ξ⟩ + 1/i)^{−1}. Then θ_{ξ} = lim_{i→∞} θ_{ξb_{i}}, and so θ_{ξ}T_{2} = lim_{i} θ_{ξ}b_{i}T_{2}. Thus, if T_{2} ∈ I_{B_{1}}(E_{2}), then θ_{ξ}T_{2}, θ_{ξ}T_{2}^{∗} ∈ K(E_{12}, E_{2}), and consequently T_{12}θ_{ξ}θ_{η}^{∗} and T_{12}^{∗}θ_{ξ}θ_{η}^{∗} are in K(E_{12}, E_{2}), which proves that T_{12} ∈ I_{K(E_{1})⊗1}(E_{12}).

If T_{2} ∈ 𝔼^{G}(B_{1}, B_{2}), then by Lemma 9 one has θ_{ξ}T_{2}x = lim_{i} θ_{ξ}b_{i}T_{2}x ≡ lim_{i} θ_{ξ}b_{i}xT_{2} = θ_{ξ}xT_{2} modulo K(E_{2}, E_{12}) for all x ∈ A(E_{2}). Hence

θ_{ξ}xT_{2} − (−1)^{∂ξ·∂T_{2}}T_{12}θ_{ξ}x ∈ K(E_{2}, E_{12})

for all x ∈ A(E_{2}). Modulo K(E_{12}), this gives

θ_{ξ}xθ_{η}^{∗}T_{12} ≡ θ_{ξ}xT_{2}θ_{η}^{∗}(−1)^{∂η·∂T_{2}} ≡ T_{12}θ_{ξ}xθ_{η}^{∗}(−1)^{∂ξ·∂T_{2}}(−1)^{∂η·∂T_{2}}
≡ T_{12}θ_{ξ}xθ_{η}^{∗}(−1)^{∂(θ_{ξ}xθ_{η})·∂T_{2}}

for all x ∈ A(E_{2}), for the last identity noticing that ∂(θ_{ξ}) = ∂ξ and ∂(x) = 0. If ∂(T_{12}) = ∂(T_{2}), this proves that [θ_{ξ}xθ_{η}^{∗}, T_{12}] ∈ K(E_{12}). If ∂(T_{12}) ≠ ∂(T_{2}) then ∂(θ_{ξ}T_{2}) ≠ ∂(T_{12}θ_{ξ}), which shows that T_{12} is a 0-connection, and in this case [θ_{ξ}xθ_{η}^{∗}, T_{12}] ∈ K(E_{12}) is obvious. Noticing θ_{ξ}xθ_{η}^{∗} = (1 ⊗ x)θ_{ξ}θ_{η}^{∗}, this shows (1).

If T_{2} ∈ 𝔼^{G}(B_{1}, B_{2}), and S_{g} = g(T_{2}) − gg^{−1}(T_{2}), then B_{1}S_{g} ⊆ K(E_{2}). By θ_{ξ} = lim_{i} θ_{ξb_{i}} we get θ_{ξ}S_{g}, θ_{ξ}S_{g}^{∗} ∈ K(E_{2}, E_{12}). Denote the G-action on E_{1} and E_{2} by U and V, respectively. Fix g ∈ G, ξ ∈ E_{1} and assume that ξ = U_{g}U_{g}^{∗}ξ. Then

θ_{ξ}g(T_{2}) − (−1)^{∂(ξ)·∂(T_{2})}g(T_{12})θ_{ξ}
= (U_{g} ⊗ V_{g})(θ_{U_{g}^{∗}ξ}T_{2} − (−1)^{∂(ξ)·∂(T_{2})}T_{12}θ_{U_{g}^{∗}ξ})V_{g}^{∗} ∈ K(E_{2}, E_{12}).

Similarly, θ_{ξ}gg^{−1}(T_{2}) − (−1)^{∂(ξ)·∂(T_{2})}gg^{−1}(T_{12})θ_{ξ} ∈ K(E_{2}, E_{12}). Hence,

(g(T_{12}) − gg^{−1}(T_{12}))θ_{ξ}θ_{η}^{∗} ∈ K(E_{12}).

This identity holds even for any ξ ∈ E_{1}, since

(g(T_{12}) − gg^{−1}(T_{12}))θ_{ξ} = (g(T_{12}) − gg^{−1}(T_{12}))θ_{U_{g}U_{g}^{∗}ξ}

for all ξ ∈ E_{1}. The identity also holds for T_{12} replaced by T_{12}^{∗}, and hence we checked (2). The claims (3) and (4) are proved similarly. (Recall Lemma 6 when checking (4).) For the last claim, it is straightforward to compute that [T_{12}^{∗}, T_{1}^{∗} ⊗ 1]θ_{ξ}θ_{η}^{∗} ∈ K(E_{12}). □

The following lemma is Lemma 2.7 of Kasparov’s paper [8].

Lemma 11. If E_{1} is countably generated and T_{2} ∈ Q_{B_{1}}(E_{2}) then there exists a T_{2}-connection T_{12} on E_{12}.

Definition 19. Let A, B_{1}, B_{2} be G-C^{∗}-algebras. An element (E_{12}, T_{12}) ∈ 𝔼^{G}(A, B_{2}) is called a Kasparov (or cap) product of (E_{1}, T_{1}) ∈ 𝔼^{G}(A, B_{1}) and (E_{2}, T_{2}) ∈ 𝔼^{G}(B_{1}, B_{2}), if E_{12} = E_{1} ⊗_{B_{1}} E_{2}, T_{12} is a T_{2}-connection on E_{12}, and a[T_{1} ⊗ 1, T_{12}]a^{∗} ≥ 0 in the quotient L(E_{12})/K(E_{12}) for all a ∈ A.

Lemma 12. Let E be a Hilbert (A, B)-bimodule, (E, F), (E, F′) ∈ 𝔼^{G}(A, B), and assume that a[F, F′]a^{∗} ≥ 0 in L(E)/K(E) for all a ∈ A. Then (E, F) and (E, F′) are operatorially homotopic.

Proof. The proof is the same as in [14], Lemma 11, and we shall only check the aspects involving G. As in Skandalis’ paper the operatorial homotopy is given by the path F_{t} = (1 + (cos t)(sin t)P)^{−1/2}((cos t)F + (sin t)F′) ∈ L(E) (t ∈ [0, π/2]) for some operator P ≥ 0 satisfying [F, F′] − P ∈ I_{A}(E). By Skandalis’ proof, (E, F_{t}) ∈ 𝔼(A, B). Given F_{1}, …, F_{n} ∈ {F, F′}, a ∈ A, and b ∈ A such that U_{g}^{∗}aU_{g} = U_{g}^{∗}U_{g}b, one has

ag(F_{1} … F_{n}) = g(bU_{g}^{∗}U_{g}F_{1} … F_{n}) ≡ g(bU_{g}^{∗}U_{g}F_{1}U_{g}^{∗}U_{g} … U_{g}^{∗}U_{g}F_{n})
= ag(F_{1}) … g(F_{n})

modulo K(E), as bF_{i} ≡ (−1)^{∂b}F_{i}b. By induction on n one gets

ag(F_{1} … F_{n}) − agg^{−1}(F_{1} … F_{n})
≡ ag(F_{1}) … g(F_{n}) − agg^{−1}(F_{1}) … gg^{−1}(F_{n}) ≡ 0

modulo K(E). Using power series, there are fixed scalars α_{n}, β_{n}, γ_{n}, δ_{n} ∈ ℂ (n ≥ 0) such that for any c ∈ A there exist K_{n} ∈ K(E) (n ≥ 0) such that

cF_{t} = ∑_{n≥0} α_{n}c(F F′)^{n}F + β_{n}c(F′F)^{n}F + γ_{n}c(F F′)^{n}F′ + δ_{n}c(F F′)^{n}F′ + K_{n}.

Note that the series is (still) absolutely convergent. In order to show that ag(F_{t}) − agg^{−1}(F_{t}) ∈ K(E), it is enough to show that ag((F F′)^{n}F) − agg^{−1}((F F′)^{n}F) ∈ K(E) for all n ≥ 0 (and similarly for the other terms). But we have checked this. □

Theorem 2. If A is separable then the Kasparov product of (E_{1}, T_{1}) ∈ 𝔼^{G}(A, B_{1}) and (E_{2}, T_{2}) ∈ 𝔼^{G}(B_{1}, B_{2}) exists and is unique up to operatorial homotopy. The Kasparov product induces a bilinear map

⊗_{B_{1}} : { \atop KK^{G}}(A, B_{1}) ⊗ { \atop KK^{G}}(B_{1}, B_{2}) → { \atop KK^{G}}(A, B_{2})

denoted by x ⊗ y ↦ x ⊗_{B_{1}} y.

Proof. Existence. By Lemma 11 there is a T_{2}-connection T̃_{2} of degree 1 on E_{12}. Put J = K(E_{12}), and A_{1} = J + K(E_{1}) ⊗ A(E_{2}). A_{1} is closed and σ-unital by [7], Section 3, Lemma 2. Note that A_{1} is G-invariant. Denote by A_{2} the C^{∗}-subalgebra (without G-action) of L(E_{12}) generated by the elements

T̃_{2} − T̃_{2}^{∗}, T̃_{2}^{2} − 1, [T̃_{2}, T_{1} ⊗ 1], [T̃_{2}, a]

for all a ∈ A. Let

Δ = {T_{1} ⊗ 1, T̃_{2}} ∪ A.

It is clear that Δ derives A_{1}, see Lemma 10. Of course, T̃_{2}−T̃_{2}^{∗} is a (T_{2}−T_{2}^{∗})- connection, and T̃_{2}^{2} − 1 is a (T_{2}^{2} − 1)-connection. Noting T_{2} − T_{2}^{∗} ∈ I_{B_{1}}(E_{2}), and writing

K(E_{1}) ⊗ A(E_{2}) = (1 ⊗ A(E_{2}))(K(E_{1}) ⊗ 1),

we get A_{1}(T̃_{2} − T̃_{2}^{∗}) ⊆ J by Lemma 10. Similarly, A_{1}(T̃_{2}^{2} − 1) ⊆ J. By Lemma 10, one has A_{1}[T̃_{2}, T_{1} ⊗ 1] ⊆ J and A_{1}[T̃_{2}, a] ⊆ J for all a ∈ A. It thus follows that A_{1}A_{2} ⊆ J. Define

φ_{1}(g) = g(T̃_{2}) − gg^{−1}(T̃_{2}), φ_{2}(g) = g(1)T̃_{2} − gg^{−1}(T̃_{2}),
φ_{3}(g) = g^{−1}(1)T̃_{2} − g^{−1}g(T̃_{2}), ψ_{1}(g) = T̃_{2}g(1) − gg^{−1}(T̃_{2}),
ψ_{2}(g) = T̃_{2}g^{−1}(1) − g^{−1}g(T̃_{2})

for all g ∈ G. We may combine φ_{1}, φ_{2}, φ_{3} and ψ_{1}, ψ_{2}, 0 to one function φ and ψ, respectively, with domain Ω being a threefold disjoint copy of G. We apply Theorem 1 to obtain M_{1}, M_{2} ∈ L(E_{12}) and set

T_{12} = M_{1}^{1/2}(T_{1} ⊗ 1) + M_{2}^{1/2}T̃_{2}.

It is well established (and straightforward to check) that (E_{12}, T_{12}) is in 𝔼(A, B_{2}) (without the set G), which is why will focus on those additional relations showing even (E_{12}, T_{12}) ∈ 𝔼^{G}(A, B_{2}). The other properties which show that T_{12} is a Kasparov product are deduced as in Skandalis [14], Theorem 12. Denote the G-action on E_{1} and E_{2} by U and V, respectively, and the diagonal action U ⊗ V on E_{12} by W. Write M̂_{i} = M_{i}^{1/2} (i = 1, 2). Since [M̂_{i}, A] ⊆ J by Corollary 1, and for any a ∈ A, W_{g}W_{g}^{∗}a = aW_{g}W_{g}^{∗} and there is a b ∈ A such that W_{g}^{∗}aW_{g} = W_{g}^{∗}W_{g}b by Definition 15,

aW_{g}M̂_{i}W_{g}^{∗} = W_{g}W_{g}^{∗}aW_{g}M̂_{i}W_{g}^{∗} = W_{g}bM̂_{i}W_{g}^{∗} ≡ W_{g}M̂_{i}W_{g}^{∗}a

modulo J for all i = 1, 2. Similarly, agg^{−1}(M̂_{i}) ≡ gg^{−1}(M̂_{i})a modulo J for all a ∈ A, i = 1, 2. Since [M̂_{i}, W_{g}^{∗}W_{g}] ∈ J, one has g(M̂_{i}T) ≡ g(M̂_{i})g(T) modulo J for any operator T ∈ L(E_{12}). Thus

ag(M̂_{1}(T_{1} ⊗ 1)) − agg^{−1}(M̂_{1}(T_{1} ⊗ 1))
≡ ag(M̂_{1})g(T_{1} ⊗ 1) − agg^{−1}(M̂_{1})gg^{−1}(T_{1} ⊗ 1)
≡ g(M̂_{1})ag(T_{1} ⊗ 1) − gg^{−1}(M̂_{1})agg^{−1}(T_{1} ⊗ 1)
≡ (g(M̂_{1}) − gg^{−1}(M̂_{1}))ag(T_{1} ⊗ 1)+
+ gg^{−1}(M̂_{1})a(g(T_{1} ⊗ 1) − gg^{−1}(T_{1} ⊗ 1))
≡ gg^{−1}(M̂_{1})gg^{−1}(a)gg^{−1}(g(T_{1} ⊗ 1) − gg^{−1}(T_{1} ⊗ 1))
≡ gg^{−1}(M̂_{1}a(g(T_{1} ⊗ 1) − gg^{−1}(T_{1} ⊗ 1))) ≡ 0

modulo J, since M̂_{1}A_{1} ⊆ J, for all a ∈ A, g ∈ G. A similar computation yields

ag(M̂_{2}T̃_{2}) − agg^{−1}(M̂_{2}T̃_{2}) ≡ agg^{−1}(M̂_{2}(g(T̃_{2}) − gg^{−1}(T̃_{2}))) ≡ 0

modulo J, since M̂_{2}φ_{1}(g) ∈ J, for all a ∈ A, g ∈ G. Thus we have proved that a(g(T_{12}) − gg^{−1}(T_{12})) ∈ J. Similar calculations show that also

(g(T_{12}) − gg^{−1}(T_{12}))a ∈ J.

Next,

ag(1)M̂_{1}(T_{1} ⊗ 1) − aM̂_{1}(T_{1} ⊗ 1)g(1) ≡ M̂_{1}(ag(1)(T_{1} ⊗ 1) − a(T_{1} ⊗ 1)g(1))
≡ 0

modulo J, since M̂_{1}A_{1} ⊆ J, for all a ∈ A, g ∈ G. Note that M̂_{2}gg^{−1}(T̃_{2}) = M̂_{2}W_{g}W_{g}^{∗}T̃_{2}W_{g}W_{g}^{∗} ≡ W_{g}W_{g}^{∗}T̃_{2}W_{g}W_{g}^{∗}M̂_{2} = gg^{−1}(T̃_{2})M̂_{2} modulo J for all g ∈ G. Since M̂_{2}φ_{2}(g) ⊆ J and ψ_{1}(g)M̂_{2} ⊆ J, one gets

ag(1)M̂_{2}T̃_{2} − aM̂_{2}T̃_{2}g(1) ≡ aM̂_{2}(g(1)T̃_{2} − gg^{−1}(T̃_{2}))
+ aM̂_{2}(gg^{−1}(T̃_{2}) − T̃_{2}g(1))
≡ a(gg^{−1}(T̃_{2}) − T̃_{2}g(1))M̂_{2} ≡ 0

modulo J for all a ∈ A, g ∈ G. It is thus evident that a(g(1)T_{12} −T_{12}g(1)) ∈ J, and by a quite similar computation, that a(g^{−1}(1)T_{12} − T_{12}g^{−1}(1)) ∈ J for all a ∈ A, g ∈ G. We have checked that (E_{12}, T_{12}) ∈ 𝔼^{G}(A, B_{2}).

Uniqueness. Consider two Kasparov products (E_{12}, F), (E_{12}, F′). In the above existence proof we defined sets A_{1}, A_{2}, Δ and Φ = {φ_{1}, φ_{2}, φ_{3}, ψ_{1}, ψ_{2}} with respect to a given T_{2}-connection T̃_{2}. To express dependence on T_{1} and T̃_{2}, let us rename this sets as A_{1}^{(T_{1},T̃_{2})}, A_{2}^{(T_{1},T̃_{2})}, Δ^{(T_{1},T̃_{2})} and Φ^{(T_{1},T̃_{2})}. Now define J = K(E_{12}), A_{1} = A_{1}^{(T_{1},F)}, A_{2} to be the C^{∗}-algebra (without Gaction) generated by A_{2}^{(T_{1},F)} ∪ A_{2}^{(T_{1},F′)} ∪ {F − F′}; Δ = Δ^{(T_{1},F)} ∪ Δ^{(T_{1},F′)}, and Φ = Φ^{(T_{1},F)} ⊔ Φ^{(T_{1},F′)}. Applying Theorem 1 with these parameters we obtain operators M_{1}, M_{2}, and we set F′′ = M_{1}^{1/2}(T_{1} ⊗ 1) + M_{2}^{1/2}F. One has F′′ ∈ 𝔼^{G}(A, B), and a[F, F′′]a^{∗} ≥ 0 and a[F′, F′′]a^{∗} ≥ 0 in L(E)/K(E). (Confer the proof in Skandalis [14], Theorem 12.) The conclusion follows by Lemma 12.

Passage to { \atop KK^{G}}. If (E_{1}, T_{1}) or (E_{2}, T_{2}) is degenerate then so is their Kasparov product. (See the proof in Skandalis [14], Theorem 12.) We have to show that the Kasparov product respects operator homotopies. Let (E_{1}, T_{1}^{t}) ∈ 𝔼^{G}(A, B_{1}) and (E_{2}, T_{2}^{t}) ∈ 𝔼^{G}(B_{1}, B_{2}) (t ∈ [0, 1]) be two operatorial homotopies. Choose a norm continuous path T̃_{2}^{t} ∈ L(E_{12}) (t ∈ [0, 1])

such that each T̃_{2}^{t} is a T_{2}^{t}-connection. Define J = K(E_{12}), A_{1} = A_{1}^{(T_{1},T̃_{2}^{0})},

A_{2} to be the C^{∗}-algebra (without G-action) generated by ⋃_{t∈[0,1]} A_{2}^{(T_{1}^{t},T̃_{2}^{t})};

Δ = ⋃_{t∈[0,1]} Δ^{(T_{1}^{t},T̃_{2}^{t})}, Ω = G × [0, 1], φ_{1}(g, t) = g(T̃_{2}^{t}) − gg^{−1}(T̃_{2}^{t}) for all (g, t) ∈ Ω, and similarly φ_{2}, φ_{3}, ψ_{1}, ψ_{2} (see above). Entering these parameters in Theorem 1 yields operators M_{1}, M_{2} and a desired operatorial homotopy (E_{12}, M_{1}^{1/2}(T_{1}^{t} ⊗ 1) + M_{2}^{1/2}T̃_{2}^{t}). □

7. The Kasparov cup-cap product

In this section all C^{∗}-algebras are assumed to be Hilbert C^{∗}-algebras. We define a slightly modified KK-theory for Hilbert C^{∗}-algebras in that we redefine Hilbert modules and equivariant maps, taking into account the structure of Hilbert C^{∗}-algebras. We denote the actions on a Hilbert C^{∗}- algebra B by g ↦ g(b) and g ↦ g^{−1}(b) for all b ∈ B, g ∈ G. All ∗- homomorphisms between Hilbert C^{∗}-algebras are assumed to be equivariant with respect to both actions g and g^{−1} (that is, we require that πg = gπ and πg^{−1} = g^{−1}π for all g ∈ G).

Definition 20. A G-Hilbert B-module E over a Hilbert C^{∗}-algebra B is a G-Hilbert B-module in the sense of Definition 11 satisfying

⟨U_{g}^{∗}x, y⟩ = g^{−1}⟨x, U_{g}y⟩, U_{g}^{∗}(xb) = U_{g}^{∗}(x)g^{−1}(b)

for all x, y ∈ E, b ∈ B, g ∈ G.

With respect to the last definition: The injectivity of g on B_{g} implies that U_{g}^{∗}U_{g} is self-adjoint by Lemma 3. This implication can be reversed, as ⟨U_{g}^{∗}U_{g}x, y⟩ = g^{−1}g⟨x, U_{g}^{∗}U_{g}y⟩ = g^{−1}g⟨U_{g}^{∗}U_{g}x, y⟩.

If A is a Hilbert C^{∗}-algebra then P = g^{−1}g is idempotent and self-adjoint, and thus the range of P is an ideal in A. This shows, for instance, that A = C[0, 1] with P (f) = g(f) = g^{−1}(f) = f(1)1 for all f ∈ A, g ∈ G is not a Hilbert C^{∗}-algebra.^{1} Actually,^{2} the elements gg^{−1} and g^{−1}g for any Hilbert C^{∗}-algebra are elements of the center of the multiplier algebra by identifying M(A) with L(A); [gg^{−1}, A] = 0 is proven in Lemma 13.

1I thank Christian Voigt for this “P is idempotent implies range(P) is an ideal?”— counterexample.

2Remarked by the referee.

Definition 21. If A is a Hilbert C^{∗}-algebra and E is a G-Hilbert B-module, then a ∗-homomorphism π : A → L(E) is called equivariant if it is equivariant in the sense of Definition 15 and

U_{g}^{∗}U_{g}π(g^{−1}(a)) = U_{g}^{∗}π(a)U_{g}

holds for all a ∈ A, g ∈ G.

Though we have now redefined G-Hilbert modules and equivariant representations for Hilbert C^{∗}-algebras, we can more or less continue with Sections 4–6 without change. Indeed, we only have to ensure that all constructions related to Hilbert modules and equivariant representations enjoy the above redefinitions, and these are only the tensor product constructions and direct sums of Hilbert modules.

Lemma 13. If A, B are Hilbert C^{∗}-algebras and π : A → B is a homomorphism then π̃ : A → L(B), π̃(a)(b) = π(a)b (a ∈ A, b ∈ B) is an equivariant homomorphism. In particular, B is a Hilbert (B, B)-bimodule.

Proof. For instance, by Lemma 3 U_{g}U_{g}^{∗} is selfadjoint and thus

U_{g}U_{g}^{∗}π̃(a)(b)=U_{g}U_{g}^{∗}(π(a)b)=(U_{g}U_{g}^{∗}(π(a^{∗})))^{∗}U_{g}U_{g}^{∗}(b)=π(a)U_{g}U_{g}^{∗}(b). □

By Lemmas 13 and 4 we may form the tensor product E ⊗_{B_{1}} B_{2} if E is a Hilbert B_{1}-module and φ : B_{1} → B_{2} a homomorphism between Hilbert C^{∗}-algebras B_{1} and B_{2}.

Lemma 14. If E_{1} is a Hilbert (A, B_{1})-bimodule, E_{2} a Hilbert (B_{2}, B_{3})- bimodule and f : B_{1} → B_{2} is a homomorphism then

π : E_{1} ⊗_{B_{1}} B_{2} ⊗_{B_{2}} E_{2} → E_{1} ⊗_{B_{1}} E_{2}, π(x_{1} ⊗ b_{2} ⊗ x_{2}) = x_{1} ⊗ f(b_{2})x_{2}

is an isomorphism of Hilbert (A, B_{3})-bimodules.

If A is unital then σ : A ⊗_{A} E → E, σ(a ⊗ x) = ax is an isomorphism of Hilbert (A, B_{1})-bimodules.

Proof. Without the G-structure this is well established. It is straightforward to compute that π and σ intertwine the G-actions. □

Definition 22. An element (E, T) ∈ 𝔼^{G}(A, B[0, 1]) generates a path t ↦ (E_{t}, T_{t}) ∈ 𝔼^{G}(A, B) (t ∈ [0, 1]) obtained by evaluation at each t ∈ [0, 1], that is, E_{t} = E ⊗_{B⊗C[0,1]} B, T_{t} = T ⊗ 1, where B ⊗ C[0, 1] → B is evaluation at time t. This path and the pair (E, T) itself will be called a homotopy between (E_{0}, T_{0}) and (E_{1}, T_{1}). The set KK^{G}(A, B) is defined as the quotient of 𝔼^{G}(A, B) by the equivalence relation given by homotopy.

Proposition 2. KK^{G}(A, B) is a quotient of { \atop KK^{G}}(A, B). KK^{G}(A, B) and

{ \atop KK^{G}}(A, B) are abelian groups with addition given by direct sum.

Proof. One proves this along the lines of [7], Section 4, Theorem 1, or [14], Proposition 4. □

Definition 23. Let A_{1}, A_{2}, B be Hilbert C^{∗}-algebras, and f : A_{1} → A_{2} a homomorphism. Then f induces a map f^{∗} : 𝔼^{G}(A_{2}, B) → 𝔼^{G}(A_{1}, B) by f^{∗}((E, T)) = (f^{∗}(E), T), where f^{∗}(E) is the Hilbert (A_{1}, B)-bimodule E with

A_{1}-action A_{1} →^{f} A_{2} → L(E). The map f^{∗} passes to the quotients KK^{G} and

{ \atop KK^{G}}, and we keep the notation f^{∗} for these maps.

Definition 24. Let A, B_{1}, B_{2} be Hilbert C^{∗}-algebras and g : B_{1} → B_{2} a homomorphism. Then g induces a map g_{∗} : 𝔼^{G}(A, B_{1}) → 𝔼^{G}(A, B_{2}) given by g_{∗}((E, T)) = (E ⊗_{B_{1}} B_{2}, T ⊗1). The map g_{∗} passes to the quotients KK^{G}

and { \atop KK^{G}}, and we keep the notation g_{∗} for these maps.

For Definition 24 one needs:

Lemma 15. Let E be a Hilbert B_{1}-module, B_{2} a Hilbert C^{∗}-algebra, and T ∈ K(E). Then T ⊗ 1, T ⊗ U_{g}U_{g}^{∗}, T ⊗ U_{g}^{∗}U_{g} ∈ K(E ⊗_{B_{1}} B_{2}) for all g ∈ G.

Proof. The proof is the same as in [7], page 523, or [5], Lemma 1.2.8, taking U_{g}U_{g}^{∗}, U_{g}^{∗}U_{g} rather than 1. □

Definition 25. Let D be a σ-unital Hilbert C^{∗}-algebra. Define

τ_{D} : 𝔼^{G}(A, B) → 𝔼^{G}(A ⊗ D, B ⊗ D)

by τ_{D}(E, T) = (E ⊗D, T ⊗1) (where E ⊗D denotes the skew tensor product).

The map τ_{D} passes to the quotients KK^{G} and { \atop KK^{G}}, and these homomorphisms are also denoted by τ_{D}.

Theorem 3. There is a Kasparov product as stated in Theorem 2, and this product also induces a bilinear map

⊗_{B_{1}} : KK^{G}(A, B_{1}) ⊗ KK^{G}(B_{1}, B_{2}) → KK^{G}(A, B_{2}).

Proof. That the Kasparov product respects homotopy may be proved in the same way as in Skandalis [14], Theorem 12. □

Proposition 3. Let A_{1}, A_{2} be separable Hilbert C^{∗}-algebras, and f : A_{1} → A_{2} and g : B_{1} → B_{2} homomorphisms.

If x ∈ KK^{G}(A_{2}, B) and y ∈ KK^{G}(B, B_{1}) (or { \atop KK^{G}}) then

f^{∗}(x) ⊗_{B} y = f^{∗}(x ⊗_{B} y).

If x ∈ KK^{G}(A_{1}, B_{1}) and y ∈ KK^{G}(B_{2}, B_{3}) (or { \atop KK^{G}}) then

g_{∗}(x) ⊗_{B_{2}} y = x ⊗_{B_{1}} g^{∗}(y).

If x ∈ KK^{G}(A_{1}, B) and y ∈ KK^{G}(B, B_{1}) (or { \atop KK^{G}}) then

g_{∗}(x ⊗_{B} y) = x ⊗_{B} g_{∗}(y).

Proof. The proof is the same as [14], Proposition 13. For the second identity one uses Lemma 14. □

Definition 26. Let A_{2}, B_{1} be σ-unital Hilbert C^{∗}-algebras and A_{1}, A_{2} be separable. The cup-cap product

⊗_{D} : KK^{G}(A_{1}, B_{1} ⊗ D) ⊗ KK^{G}(D ⊗ A_{2}, B_{2}) → KK^{G}(A_{1} ⊗ A_{2}, B_{1} ⊗ B_{2})

is defined by the formula x_{1}⊗_{D}x_{2} = τ_{A_{2}}(x_{1})⊗_{B_{1}⊗D⊗A_{2}} τ_{B_{1}}(x_{2}). The cup-cap

product for { \atop KK^{G}} is defined in the same way.

Lemma 16. If x ∈ KK^{G}(A, B) (or { \atop KK^{G}}(A, B)) and f : A′ → A and g : B → B′ are homomorphisms, then τ_{D}(f^{∗}(x)) = (f ⊗ 1)^{∗}(τ_{D}(x)) and τ_{D}(g_{∗}(x)) = (g ⊗ 1)_{∗}(τ_{D}(x)).

Proof. Let x = (E, T). For the second claim we use

(E ⊗ D) ⊗_{B⊗D} (B′ ⊗ D) ≅ (E ⊗_{B} B′) ⊗ (D ⊗_{D} D)

(see Kasparov [7], Section 2, page 523). □

Lemma 17. If x ∈ KK^{G}(A, B) and f : D_{1} → D_{2} is a homomorphism then (1 ⊗ f)^{∗}(τ_{D_{2}}(x)) = (1 ⊗ f)_{∗}(τ_{D_{1}}(x)).

Proof. One checks that the proof of Skandalis [14], Lemma 7, works also in our setting. □

Proposition 4. Let B_{1}, B′_{1}, B′_{2}, D′ be σ-unital Hilbert C^{∗}-algebras and A_{1}, A′_{1}, A_{2}, A′_{2} be separable. Let f_{1} : A′_{1} → A_{1}, f_{2} : A′_{2} → A_{2}, g_{1} : B_{1} → B′_{1}, g_{2} : B_{2} → B′_{2}, h : D → D′ be homomorphisms. Then the cup-cap product of Definition 26 satisfies

f_{1}^{∗}(g_{1} ⊗ 1)_{∗}(x_{1}) ⊗_{D} (1 ⊗ f_{2})^{∗}g_{2}_{∗}(x_{2}) = (f_{1} ⊗ f_{2})^{∗}(g_{1} ⊗ g_{2})_{∗}(x_{1} ⊗_{D} x_{2}),
(h ⊗ 1)_{∗}(x_{1}) ⊗_{D′} x_{2} = x_{1} ⊗_{D} (h ⊗ 1)^{∗}(x_{2}),

with the restriction that if f_{2} or g_{1} are not trivial (i.e., are not the identity map) then this only holds in KK^{G}.

Proof. This is some computation by applying the formulas of Proposition 3 and Lemmas 16 and 17. □

Let 1 ∈ { \atop KK^{G}}(ℂ, ℂ) (or KK^{G}(ℂ, ℂ)) be given by the Hilbert (ℂ, ℂ)- bimodule ℂ with trivial grading and action, and the zero operator.

Proposition 5. Let A be separable and x ∈ KK^{G}(A, B) (or { \atop KK^{G}}(A, B)). Then x⊗_{ℂ}1 = x. If A is unital and g(1_{A}) = 1_{A} for all g ∈ G then 1⊗_{ℂ}x = x.

Proof. One proves this along the lines of [14], Proposition 17. □

Theorem 4. Suppose that A is a separable and B a σ-unital Hilbert C^{∗}- algebra. Then the map { \atop KK^{G}}(A, B) → KK^{G}(A, B) is an isomorphism.

Proof. The proof is the same as Theorem 19 of Skandalis [14]. □

530 Bernhard Burgstaller

Theorem 5. Assume that B_{1}, B_{2} are σ-unital Hilbert C^{∗}-algebras, A_{1}, A_{2}, A_{3}, D_{1} are separable Hilbert C^{∗}-algebras, and

x_{1} ∈ KK^{G}(A_{1}, B_{1} ⊗ D_{1}),
x_{2} ∈ KK^{G}(D_{1} ⊗ A_{2}, B_{2} ⊗ D_{2}),
x_{3} ∈ KK^{G}(D_{2} ⊗ A_{3}, B_{3}).

Then

(x_{1} ⊗_{D_{1}} x_{2}) ⊗_{D_{2}} x_{3} = x_{1} ⊗_{D_{1}} (x_{2} ⊗_{D_{2}} x_{3}).

Proof. One proves this along the lines of [14], Theorem 21. □

Acknowledgement. I thank Radu Purice and Florin Radulescu for their invitation and hospitality at the Institute of Mathematics of the Romanian Academy (IMAR) in Bucharest, where the main part of this work was developed.

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Doppler Institute for mathematical physics, Trojanova 13, 12000 Prague, Czech Republic

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