KK-theory of reduced free product C -algebras Emmanuel Germain UFR de Mathematique Universite Denis Diderot Paris 7 755 Paris, France germain@mathp7.jussieu.fr (Accepted Duke Math Journal April 1995) Introduction For unital C -algebras endowed with states there is a natural reduced free product construction which generalizes the C -algebra of the regular representation of a free product group. Whereas the question of computing the K-theory and all the various KK-groups is completly understood in the case of discrete groups (see the work of Pimsner in [13]), little is known so far for more general situations. In this paper, we prove an analogue of Cuntz's K-amenability result (see [3]) showing that there is a K-theoretical equivalence between the reduced and full free product of nuclear C -algebras. This shows in particular that the reduced free products obtained for dierent choices of states (with a natural restriction) are all K-equivalent. Since it is often easier to compute the K-theory of the full free product of C -algebras, our result is an important step in our way to understand the reduced free product C -algebra at a K-theoretical level. Furthermore the tools developed here allow us to give a unied treatment and to extend to a larger set of C -algebras the computation of the KKgroups of full free product C -algebras. To this purpose we introduce the notion of K-pointed C -algebras. This is the K-theoretic generalization of a unital C -algebra endowed with a one-dimensional representation. For those algebras we prove the exact sequences conjectured by Cuntz in [4] by an argument that generalizes his proof for C -algebras endowed with a one-dimensional representation. 1
This work is organized as follows. The rst two sections consist of preliminaries needed for the constructions of section 3 and 4. The rst section contains a few lemmas about the homogeneous space of nite sets of orthonormal vectors in the canonical stable A-Hilbert module. As for Hilbert spaces, we show that unitaries which are compact perturbations of the identity act transitively on these sets. In section 2 we recall technical nuclearity results for Kasparov bimodules. As shown in [16], for any faithful representation of a separable C -algebra A, there is a presentation of the identity Kasparov bimodule for A by a bimodule with this representation as left action. Moreover if A is nuclear, we can also construct a homotopy between this module and the canonical identity bimodule which in some sense `lies' in the given representation. Let A r denote now the reduced free product C -algebra of a set of K- nuclear C -algebras endowed with states, and A the full free product C - algebra. The tasks assigned to the third section is to exhibit a candidate for the inverse in KK(A r ; A) of the canonical morphism from A to A r and to show that this morphism always realizes a K-theoretical sub-equivalence (see [17] for denitions). We get this result by a certain free product of homotopies of the previous section. Finally in section 4 we show how it is possible to prove that in the case of nuclear C -algebras we actually have a full K-theoretical equivalence. It relies on a crucial proposition that states when a particular deformation of the reduced free product representation still factorizes through the reduced free product C -algebra. The following section develops then the main properties and examples of K-pointed C -algebras and ends by the computation of the K-groups of their full free product. We nish with some additional applications and remarks. The results of this paper are part of the author's doctoral dissertation at U.C. Berkeley. The author wishes to thank D. Voiculescu for his helpful comments and his encouragments during the preparation of this work and G. Skandalis for his reading of preliminary versions and his many suggestions on how to improve the original proofs. Throughout this article, we shall only consider unital separable C -algebras. Also all free products are taken in the category of unital algebras. 2
1 The homogeneous space of A-orthonormal sets Let A be a separable C -algebra with unit. Let H be an innite dimensional separable Hilbert space, and H P A the canonical A-Hilbert module. I.e. the Hilbert module of all sums i i a i with i in H and a i in A such that P i;j < i ; j > a i a j converges in A: Let F n;1 be the set of all A-orthonormal sets of n vectors in H A: Let U(H A) denote the group of unitaries of L(H A): See [17] for the appropriate denitions of these spaces of morphisms. Lemma 1.1 U(H A) acts transitively on F n;1. Proof: Identify rst H with `2(N): Let ( 1 ; ::; n ) be in F n;1 and (e 1 ; ::; e n ) be an orthonormal set in H. We only need to prove that there exists a unitary that sends e i 1 A on i. For ; 2 H A, dene ; to be the following compact operator ; (x) = < ; x > A : Let p = P n i=1 i ; i. We must show that (1? p)`2(n) A is isomorphic to `2(N) A: But according to [8] x6 Lemma 5, there exists a m 2 N and a unitary u such that upu :`2(N) A `2(f1; ::; mg) A. So u(1? p)`2(n) A = `2(N? f1; ::; mg) A `2(f1; ::; mg) A \ u(1? p)`2(n) A: So by Kasparov stabilization theorem u(1?p)`2(n)a is isomorphic to `2(N)A: Dene now a unitary V from H A = ( W n i=1 e i ) A ( W n i=1 e i )? A to p(h A) (1? p)h A by V:e i 1 = i and V realizing the identication of ( W n i=1 e i)? A to (1? p)h A as proved above. Corollary 1.2 connected. Since U(H A) is contractible (see [12]), F n;1 is Let U 1 be the group of unitaries of C1 + K(H A): And let (e 1 ; ::; e n ) be an orthonormal set of H. Dene from U 1 to F n;1 by (g) = (g(e 1 1 A ); ::; g(e n 1 A )): Lemma 1.3 i. has the homotopy lifting property with respect to any compact set. ii. The image under of the connected component of the identity in U 1 is F n;1 : Proof of i. 3
We need to prove that for all homotopy H : X [; 1]?! F n;1 (x; t) 7?! ( i (x; t)) with X compact and g : X! U 1 continuous such that (g (x)) = H((x; )); for all x 2 X there exists e H : X [; 1]! U1 continuous with ( e H) = H and e H((x; )) = g (x); for all x 2 X: By replacing A by A C(X), we only need to prove i. for X a point. We can also assume that g = 1. Let p t = P i i (t); i (t) be the projection associated to the orthonormal set. Recall now proposition 4.3.3 in [2] : Proposition If (p t ) t2[;1] is a continuous path of projections of a unital C -algebra B, then there exists a continuous path of unitaries u t of B such that u = 1; 8t 2 [; 1]; u t :p u t = p t: By the proposition there exists a continuous path in U 1 such that u = 1; u t : X i ei 1;e i 1:u t = X i i (t); i (t): Therefore u t : i (t) is a orthonormal basis of ( W n i=1 e i ) A: Dene then unitaries v t 2 U 1, by v t :e i 1 = u t : i (t) and v t = Id on ( W n i=1 e i )? A, then eh(t) = v t :u t is the needed homotopy. Proof of ii. By corollary 1.2 F n;1 is connected. Let then f i (t)g be a path in F n;1 between fe i 1 A g and the considered orthonormal set. Then i. when X is a point gives the result. Corollary 1.4 : F n;1 is contractible for all n: Proof: By 1.3.i we know that the obvious map from U(H A) to F n;1 is a weak bration (see [18]). Since both ber and total space are isomorphic to U(H A), hence contractible, all homotopy groups for F n;1 vanish. But since it is a CW-complex this means that F n;1 is contractible. 2 Nuclear Kasparov bimodules Let's recall the denition of a nuclear bimodule (as given in [16] denition 1.6) Definition : Let A; B be two unital separable C -algebras and E a countably generated right B-Hilbert module with a left action of A. E is a 4
nuclear A; B-bimodule if for every integer n and every vectors 1 ; :::; n in E the map from A to M n (B) given by a 7! (< i ; a: j >) i;j is nuclear. Remark : If is a representation of A in a Hilbert space H then 1 B : A! H B is K-nuclear. For A; B trivially graded unital separable C -algebras a cycle (or Kasparov bimodule) in KK(A; B) is a pair (E; F) with E = E + E? a Z 2 -graded bimodule (the action of A and B are unital and of degree ) and F 2 L(E) of degree 1 such that F = F, F 2? 1 2 K(E), 8a 2 A; [(a); F ] 2 K(E) where denote the left action of A: A cycle is degenerate if moreover F 2 = 1 and F commutes with the action of A: Definition A is said to be K-nuclear if the identity bimodule is K- homotopic to a nuclear Kasparov bimodule. Let be a representation of A in H such that?1 (K(H)) =. And note L A the left multiplication in A: We need to prove a slightly more precise proposition, namely Proposition 2.1 If A is K-nuclear then there exists a unitary U 2 L(H A A; H A) 1 and a A-A C([; 1]) Kasparov bimodule ((H A C([; 1])) 2 ; ) 1 for which the evaluation at t = 1 is a degenerate Kasparov bimodule and the evaluation at t = has U( 1 A L A )U 1 A as left action of A: Proof By proposition 2.6 a. of [16] we already know that there exists a A-A C([; 1]) Kasparov bimodule ((H A C([; 1])) 2 ; F ) for which the evaluation at t = 1 is a degenerate Kasparov bimodule and the evaluation at t = has an action of A of the desired form. We only have to prove that F is a compact perturbation of a unitary to be done. Since F is degenerate at t = 1, it is therefore a degenerate element of K (K(H) A C([; 1])) by homopoty invariance. Now because of Corollary 2 of [12], this proves that F actually is a compact perturbation of a unitary. Corollary 2.2 Let be a distinguished unit vector in H: In the above proposition we can then assume that U( 1 A ) = 1 A : By 1.3 ii. above we know that there exists a continuous path u t Proof in U 1 such that u 1 = 1 and u sends U( 1 A ) to 1 A : We can thus conjugate the action of A by the unitary of L(H A C([; 1])) that this path denes. We still get a Kasparov bimodule (since we have a path in U 1 ) and it still degenerates at t = 1: 5
Remark 2.3 Let S be the restriction of U to H A then (H 2 A; S 1 A 1 A ; S ) is KK-equivalent to the identity bimodule. And S is an isometry of index 1 with KerS = A: Let's view C([; 1)), the continuous functions vanishing at t = 1, as a C([; 1]) Hilbert module. We are now ready for a further renement of proposition 2.1 when the algebra A is nuclear. Proposition 2.4 If A is nuclear then there exists a unitary U 2 L(H A C([; 1]) A C([; 1)); H A C([; 1])) such that ((H 1 A C([; 1])) 2 ; ) with the following left action of A 1 e t (a) = U((a) 1 AC([;1]) 1 C([;1)) L A (a))u (a) 1 AC([;1]) is a A-A C([; 1]) Kasparov bimodule for which the evaluation at t = 1 is degenerate. Proof Since A is nuclear, L A is nuclear as in the denition above, therefore we can invoke theorem 1.5 in [16] Theorem There exists an isometry v 2 L(`2(N) A; H A) such that, for all a 2 A, 1L A (a)?v ((a)1 A )v is a compact operator of `2(N)A: This theorem has the following as immediate corollary Corollary There exists a unitary W 2 L(H A `2(N) A; H A) such that W ((a) 1 A 1 L(a))W? ((a) 1 A ) 2 K(H A): Proof of the corollary Let p = vv be the range projection of v, and (a) = (1? p)((a) 1 A )(1? p) a completly positive map. Then V = 1 v from (1? p)(h A) `2(N) A to H A is a unitary that intertwines 1 L A and 1 A modulo the compact operators since p((a) 1 A )(1? p) is compact for all a 2 A (see x7 remark 2 in [7]). Thus modulo compact operators, 1 A 1 L A is unitarily equivalent to 1 L A 1 L A. But since `2(N N) is isomorphic to `2(N), 1 L A 1 L A is unitarily equivalent to 1 L A, which is unitarily equivalent modulo the compacts to 1 A. Set now W = W 1 C([;1]) : By Kasparov stabilization theorem (see [7]) there exists a unitary W 2 L(`2(N)C([; 1]C([; 1)); `2(N)C([; 1])): 6
If we consider the following unitary V = W (1 HA W 1 A )(W 1 AC([;1)) ) then it is clear that V intertwines (a) 1 AC([;1]) 1 C([;1)) L A (a) and (a) 1 AC([;1]) modulo compact operators. Hence the evaluation at t = 1 yields a unitary V 1 which intertwines modulo compact operator 1 A with itself. Therefore U = (V 1 1 C([;1]) )V is the desired unitary. We will continue to denote U t the evaluation at t of the unitary U: Corollary 2.5 Let 2 H denote an orthonornal vector, orthogonal to : We can arrange that the unitary of proposition 2.4 satises the conditions: i. For t 2 [; 1=2], 8 >< >: U t ( 1 A ) = cos(t) 1 A + + sin(t) 1 A U t ( 1 A ) =? sin(t) 1 A + + cos(t) 1 A ii. For t 2 [1=2; 1], U t ( 1 A ) = 1 A Proof For t 2 [3=4; 1], do as corollary 2.2 to get a U that leaves 1 A xed. For t 2 [1=2; 3=4] thanks to lemma 1.3 ii. we can choose a continuous path of unitaries (V t ) t2[1=2;3=4] in C1 K((H? C ) A) such that V 3=4 = 1 and V 1=2 U 3=4 :1 A = 1 A : So we let's take U t = (V t 1 A):U 3=4 : For t 2 [; 1=2] dene (W t ) t2[;1=2] by W t : 1 A = cos(t) 1 A + sin(t) 1 A W t : 1 A =? sin(t) 1 A + cos(t) 1 A W t = 1 on (C C)? A: and let's take U t = W t U 1=2 : 3 An `inverse' in K-theory 3.1 Preliminaries and notations Part of this material is extracted from [2] x1. 7
of Hilbert spaces with distin- Definiton 3.1.1 Let I be an at most countable index set. The free product of a family (H i ; ) i2i guished normal vector, is (H; ) with where M M H = C (H i 1 H i p ) p1 i2d p D p = fi = (i 1 ; :::; i p ); i j 2 I; i j 6= i j+1 for j 2 f; :::; p? 1gg and for any subspace K containing, K is the ortho-complement of C in K: Proposition 3.1.2 For k 2 I, let's dene two Hilbert spaces H(; k) and H(; k) by H(; k) = C M p1 H(; k) = C M p1 M i2d p i 1 6=k M i2d p i p6=k (H i 1 H i p ); (H i 1 H i p ): Consider also the unitaries V k : H! H k H(; k) and W k : H! H(; k) H k dened by V k (h 1 h p ) = W k (h 1 h p ) = 8 >< >: 8 >< >: where h j 2 H i j and (i 1 ; :::; i p ) 2 D p : h 1 (h 2 h p ) if i 1 = k; p 2 h 1 if i 1 = k; p = 1 (h 1 h p ) if i 1 6= k (h 1 h p?1 ) h p if i p = k; p 2 h 1 if i 1 = k; p = 1 (h 1 h p ) if i p 6= k From this, from any B-Hilbert module E we can construct two representations of L(E H k ) resp. L(H k E) into L(E H) resp. L(H E) E k(a) = (1 E V k ) (a 1 H(;k) )(1 E V k ) 8
E k(a) = (W k 1 E ) (1 H(;k) a)(w k 1 E ): The following lemma is important Lemma 3.1.3 Let a 2 L(H k ) and b 2 L(H l E) then [ k (a) 1 E ; E l (b)] = kl (P k 1 E ) E k([a 1 E ; b]) where P k is the projection of H onto H k = C: H k from the decomposition of H: Notation 3.1.4 If B is a C -algebra, E a B-Hilbert module and a representation of B into L(F) where F is a D-Hilbert module, then we can dene a D-Hilbert module E F completion of the algebraic tensor product for the D-valued inner product: < 1 1 ; 2 2 >=< 1 ; (< 1 ; 2 > B ): 2 > D and a map from L(E) to L(E F) by T 7! T 1 F : Properties 3.1.5 1. If is faithful then the map T 7! T 1 F is isometric 2. If E = B and is non degenerate (i.e. (B):F = F) then E F ' F: Notation 3.1.6 If E = H k B with B unital and a unital homomorphism between B and D then E D ' H k B D ' H k D and we will note somewhat improperly (because L(H B) 6= L(H) B)) k the map from L(H k B) into L(H D) dened by k (a) = D k (a 1) = ( B k (a)) 1: Let dene now the reduced free product of a family (A i ; ' i ) i2i of unital C -algebras with specied states. Definiton 3.1.7 If (H 'i ; 'i ; ) is the GNS construction for A i, then A r =? n i= (A i; ' i ) is the C -algebra generated by [ n i=1 i( 'i (A i )) in the free product Hilbert space? n i= (H ' i ; ) and will denote the canonical map of the full free product into the reduced one. Remark 3.1.8 Let x be in L(K A i ) with K a Hilbert space then (x ji 1 A ) 1 H = K i (x i 1 Hi ): 9
3.2 The sub-k-equivalence Let (A i ) i2i be an indexed set of unital separable C -algebras with ' i as states. (H 'i ; 'i ; ) the GNS construction for each A i ; and assume that 'i is faithful for all i: Note A their full free product, j i the natural inclusion of A i in A and L i (resp. L) the left multiplication in A i (resp. A). Let H 1 be C C 1 and H i = (H 1 ; )? (H 'i ; ); i = i ( 'i ): In fact i is the sum of innitely many copies of 'i, thus?1 i (K(H i )) = since 'i is faithful. We will call again the vacuum vector of H i ; H will denote the free product Hilbert space with respect to of the H i ; and the free product representation? i2i i ( i ): Remark 3.2.1 By associativity and commutativity of the free product of Hilbert spaces we have H = (? i2i H 1 )? (? i2i (H 'i ; )): Thus = 2 (? i2i i ( 'i )); therefore factorizes through : From now on let's assume that each A i is K-nuclear. Theorem 3.2.2 : There exists a homotopy in KK(A; A C([; 1])) between a degenerate element and the identity K-module plus a Kasparovmodule of the form (H A `2(I) H A; 1 A 1`2 (I) 1 A ; ) with S an S S isometry of codimension 1: Let's give the following important corollary to this theorem Corollary 3.2.3 If (A i ; ' i ) is an at most countable set of unital separable K-nuclear C -algebras with states such that the associated GNS representations are faithful, then the canonical map of the full free product into the reduced one is a K-theoretical sub-equivalence. Proof of corollary: By remark 3.2.1 we can now easily dene an 2 KK(A r ; A) such that = Ar : Therefore Ar + 1 A = in KK(A; A) by theorem 3.2.2. Proof of 3.2.2 For every i consider the homotopy arising from corollary 2.2 applied to A i and i : Call e i = e + i e? i the representation of A i and U i the unitary appearing there. Call also i (x) the restriction of i (x) to the stable subspace H i H(; i) : Then e + i ji 1 A ( i i ) 1 AC([;1]) and 1
e? i ji 1 A ( i i ) 1 AC([;1]) are two unital representations of A i into H A C([; 1]) which are equal modulo the compact operators. Then e + =? i2i (e + i ji 1 ( i i ) 1) e? =? i2i (e? i ji 1 ( i i ) 1) are representations of A into L(H A C([; 1])) that are equal modulo compact operators, thus dening an element of KK(A; A C([; 1])) which is degenerated at t = 1 and at t = we have the two representations:? = 1 A and + =? i2i (U i ( i 1 Ai L i )(U i ) ) ji 1 A ( i i ) 1): The following lemma will conclude the proof: Lemma 3.2.3 There exists a unitary V 2 L(`2(I) H A A; H A) such that + = V (1`2 (I) 1 A L)V : Proof Decompose each U i as S i P i according to the decomposition of L(H i A i A i ; H i A i ) into L(H i A i ) L(A i ; H i A i ). Recall that P i is the isometry that sends A i onto A i : Then V = M i2i i j i (S i ) P is a unitary where P = P i ji 1 A (it does not depend on i). Indeed it is a sum of isometries with orthogonal range whose sum is H A (since R( i j i (S i )) = H(; i) H i A.) Since our situation is clearly invariant by permutation we only have to prove that for a particular i 2 I and for all a 2 A i, + (a) = V (1`2 (I) i ( i (a)) 1 A L i (a) 1 ji A)V : But this is clear on H i A H A since V reduces to (S i P i ) 1 ji A and + (a) to (U i ( i 1 L Ai i )(U i ) ) 1 ji A: On the other hand when restricted to H(; i ) H i A; V reduces to A i (S i 1 ji A), + (a) to i ( i (a)) 1 A : Lemma 3.1.3 gives that on the considered subspace A i (S i ji 1 A)( i ( i (a)) 1 A ) A i (S i ji 1 A) = i ( i (a)) 1 A since A i (S i ji 1 A):H(; i ) H i A? H i A: Lastly for each i 6= i, on H(; i) H i A, V reduces to A i (S i j i 1 A ) and + (a) to i ( i (a))1 A ; therefore, as above, on this subspace + (a) = V (1`2 (I) i ( i (a)) 1 A L i (a) ji 1 A)V : 11
Since the sum of all the subspaces considered is H A, we proved the lemma. Corollary 3.2.4 The free product of any pair of unital separable K- nuclear C -algebras is K-nuclear. Indeed since they are separable, they have a faithful state. Theorem 3.2.2 gives the result since the identity bimodule is K-equivalent to the nuclear bimodule (H A `2(I) H A; 1 A 1`2 (I) 1 A ; S = i A i (S i ji 1 A ): 4 The case of nuclear C -algebras S S ) with Let I be a countable index set, and (A i ) i2i unital separable nuclear C - algebras with states ' i : We continue to work with the notations introduced in x3.2, in particular i is innitely many copies of the GNS representation associated to ' i (which we suppose faithful) and so is =? i2i i () compared to? i2i i ( 'i ): H is still the free product of the H i : The whole section is devoted to prove the following Theorem 4.1 With the above algebras the canonical map form A to A r is a K-equivalence. Thanks to corollary 3.2.3, it is enough to prove that Ar = 1 Ar : For each i choose a unit vector i 2 H i orthogonal to and for each algebra A i consider the homotopy of corollary 2.5. Following the proof of theorem 3.2.2 we can then construct a homotopy h in KK(A; A C([; 1])). It is clear that we only have to show that h 1 1 ArC([;1]) factorizes through A r for it yields a homotopy in KK(A r ; A r ) between a degenerate element and 1 Ar + A : That is to say since e? = (? i2i i ) 1 AC([;1]) and : A r! L(H ' ) is faithfull that e + 1 1 HC([;1]) as a representation of A into L(H H C([; 1])) factorizes through A r. Because for an x 2 L(H H C([; 1])) we have that kxk = sup t kx t k it is enough to prove that for all t e + t 1 H = [? i2i ( t i ji 1 A ( i i ) 1 A )] 1 H where t i = (U i t)( i 1 Ai L i (a))(u i t) and i (x) is the restriction of i (x) to H i H(; i), factorizes through A r : We can rewrite the representation 12
as e + t 1 H =? i2i ( H i i ( t i i 1 Hi ) ( i i ) 1 H ) using remark 3.1.8. Because of corollary 2.5, we will be done if we prove the following Proposition 4.2 For each i 2 I let K i be a Hilbert space, and i be a representation of A i into L(H i H i ) such that there exists a unitary U i 2 L(H i K i ; H i H i ) with the properties 1. i = U i ( i 1 Ki )U i 2. U i ( ) 2 K i Then the representation =? i2i ( H i i ( i ) ( i i ) 1 H ) of A into L(H H) is an innite direct sum of? i2i i ( i ): Proof : Consider the following subspaces: L k = U k ( K i ) H k H k L k C = L k E k = L k H(; k) H k H E k = L k H(; k) H k H E k = H(; k) H k H F k = H(; k) H H H Note that E k = E k E k C and E k? E l for l 6= k: Let E = k E k C : What we will prove is i. 8k 1, 8(i 1 ; :::; i k ) 2 D k, 8x ij 2 Ker ' ij A ij, 8s; s 2 E, we have < (x i1 x ik ):s; s >= ii. (A):E = H H 13
Because of the nature of the reduced free product representation (see [2] x1) this will prove the proposition. We settle i. by induction thanks to the lemma: Lemma 4.3 Let T 2 (j k (x k )) for x k 2 Ker ' k : Then we have for all l 6= k T:(E k F k ) (F l E l ) \ E? Proof : First T (F k ) H k H(; k) H, thus T (F k ) F l \ E? for l 6= k: Now T (E k ) U k (H k K i) H(; k), so T (E k )? F k and by unitarity of U k, T (E k )? E k : Since E l F k E k for all l 6= k, we have T (E k)? E: Also since? T (E k ), T (E k ) H k H(; k) H k H E l F l for l 6= k: Finally let's prove ii. Let V = (A):E Let's remark that on k H k H(; k) H; (x l) = l ( l (x l )) for x l 2 A l and k 6= l: P Thus an easy induction shows that we only need to prove that V contains ( k H k) H: Let (H n ) n be the obvious decomposition of H by the length of the tensors ( for n 6=, H n = L i2d n (H i 1 H i p ) and H = C ). We now show by induction that V contains ( P k H k H k (H(; k)\h n ): The case n = is obvious since V contains P k L k and (j k (A k )):L k = H k H k : Assuming the case n? 1, we deduce that (H(; k) \ H n ) is in V and since E k is in V, we conclude that L k (H(; k) \ H n ) is in V for all k, therefore so is H k H k (H(; k) \ H n ): Remark 4.4 Following the proof given here one can see that there is a more technical condition that insures K-equivalence: we only need to know that there exists for all i a homotopy (e + i ; e? i ) such that, for all t and with e t the evaluation at t, e + i Ai i C([;1]) e t is conjugated as a representation of A i to a sum of i 's (and the same should hold for e? ). i 5 K-pointed C -algebras Definition 5.1 A unital C -algebra A is said to be K-pointed if there 14
exists 2 KK(A; C) such that i A () = 1 C with i A the inclusion of C in A given by the unit. As an easy consequence of Proposition 2.6 a. of [16] ( and since C is nuclear) we have Proposition 5.2 Let : A! L(H) be a representation that does not intersect the compact operators, then for A K-pointed there exists an isometry V A of index 1 in L(H) that have compact commutator with (a) VA for all a 2 A and such that is represented by (H 2 ; ; V ) A Here is a list of easy properties Proposition 5.3 i. Any unital algebra with a representation of dimension 1 is K-pointed (or any pointed C -algebras is K-pointed). ii. If A and B are two unital C -alegbras and u an element of KK(A; B) is such that i A A u = i B (with i A and i B the inclusion of C in A and B respectively) then B K-pointed implies A K-pointed. Proof If B is an element of KK(B; C) such that i B B B = 1C, then A = u B B veries i A A A = 1C: iii. In particular for every K-amenable discrete group, the reduced C - algebra of the group is K-pointed Proof In [6] proposition 3.4 Julg and Valette proved that in such a case the canonical homomorphism from the full group C -algebra to the reduced one is a K-equivalence. Hence there exists an element u in KK(C r(g); C (G)) which statises the hypothesis of ii. Recall also that the full group C -algebra is K-pointed by i. iv. If A and B are K-pointed then so are A max B and A min B The Kasparov tensor product over C of A and B gives the Proof correct element of KK(A max B; C); resp. of KK(A min B; C): v. If A and B are K-pointed then so is A? B and A? red B ( for the latter case the GNS representations associated to the states need to be faithful). In particular the extension by the compacts of the Cuntz algebra O n is K-pointed, since it is a n-fold reduced free product of Toplitz algebras (see x2 of [2]). 15
Obvious using proposition 5.2 above and the construction of lemma 1.13 of [2]. Proof vi. If A is K-pointed by A and is an automorphism of A such that ( A ) = A (in particular if is homotopic to the identity) then A Z is K-pointed Proof The Pimsner-Voiculescu exact sequence in K-homology reads KK(A; C) 1?? KK(A; C) j A?KK(A Z; C) with j A the natural (unital) homomorphism of A in A Z: Since (1? )( A ) = by exactness there exists an in KK(A Z; C) such that j A = A : In particular A ; the `non-commutative' 2-torus is K-pointed. vii. If A is K-pointed then i A from Z = K (C) to K (A) has a left inverse (i.e. 1 A is of innite order and K (A) = Z:1 A N with N a sub-group.) Conversely, if A satises the universal coecients formula (see [15]), (i.e. A is K-equivalent to an abelian algebra) then the existence of a left inverse for i A implies that A is K-pointed. Proof Let be the natural map from KK(A; B) to Hom(K (A); K (B)): Then if A is K-pointed (by A ) we have K (A) = Z:1 A Ker(( A )): Moreover if f is a left inverse for i A there exists A such that ( A ) = f (because is onto). By naturality and since is an isomorphism when A = B = C we have i A A A = 1C: Remark 5.4 M n (C) is not K-pointed unless n = 1: Indeed the (unital) inclusion of C in M n (C) is equal, in KK(C; M n (C)), to n times the homomorphism that gives the Morita equivalence. Let's now prove a theorem about free product of K-pointed C -algebras. Theorem 5.5 Let A and A 1 be two K-pointed C -algebras. Then A A 1 is K-equivalent to A? A 1 C: Proof Consider j and j 1 the natural inclusion of A (resp. A 1 ) in A? A 1 ; i and i 1 the inclusion of C in A (resp. A 1 ) given by the units. Let also p be the canonical map from A? A 1 to A max A 1 : We have the following lemma 16
Lemma 5.6 If A and A 1 are K-pointed (by and 1 respectively) then there exists 2 KK(A? A 1 ; A ) and 1 2 KK(A? A 1 ; A 1 ) such that 8 >< >: j A?A 1 = 1 A j 1 A?A 1 1 = 1 A1 j A?A 1 1 = C i 1 j 1 A?A 1 = 1 C i A = 1 A1 1 A j + 1 A1 j 1 = 1 A?A 1 + ( A ) C i A?A 1 with i A?A 1 = j i = j 1 i 1 : Proof of lemma 5.6: Let k = p ( Ak ( 1?k )) for k = or 1 with D : KK(A; B)! KK(A D; B D) the external tensor product (see [17]). The rst four assertions are easy and left to the reader. The last two are a little more technical. By denition A = p ( A ( 1 ) ): Hence A = p () where 2 KK(A A 1 ; C) is the Kasparov tensor product over C of the elements and 1 : By proposition 6.4 of [17] this product is commutative, therefore the order does not intervene, thus we have the fth identity. Now let's suppose that is represented by the cycle (H 2 ; ; F ): Then k is represented by (H 2 ; k k ; F ) where k = p j k : To achieve this goal we must show that (p j k ) () = k : But since p j k = k (i 1?k ) where k stands for Ak, we have (p j k ) A A 1 = k (i 1?k ) A A 1 k ( 1?k ) Ak k = k (i 1?k A1?k 1?k ) Ak k = k (1C) Ak k = 1 Ak Ak k = k We are now ready to compute the right hand side of the last identity: A j + 1 A1 j 1 is given by denition by the cycle ((H H) 2 A? A 1 ; e; F 1 A?A 1 F 1 A?A 1 ) with e the free product of (a e (a ) = ) 1 A?A 1 1 H j (a ) 1H j e 1 (a 1 ) = 1 (a 1 ) 1 (a 1 ) 1 A?A 1 17
There exists a homotopy that moves e 1 into 1 (a 1 (a 1 ) = 1 ) 1 A?A 1 1 H j 1 (a 1 ) by conjugating the representation by a path a unitaries in M 2 (C) from 1 the identity to. Since the free product morphism j 1? j 1 is the identity of A? A 1 and? 1 is p we have A j + 1 A1 j 1 = (1 A?A 1 ) (i A A 1 ()) + p () i A?A 1 where i A A 1 = p i A?A 1 : By Proposition 5.3 v. i A A 1 () = 1C, therefore we have proved the last assertion of the lemma. Proof of theorem 5.5: The element P 2 KK(A A 1 ; A?A 1 C) given by j j 1 and Q 2 KK(A? A 1 C; A A 1 ) given by i (?i 1 ) ( A (1 A? i )) 1 realize the K-equivalence. This easy check is left to the reader. Corollary 5.7 If (A i ) i2f;1g are unital separable K-pointed C -algebras, then for any separable C -algebra D we have and! KK(D; C) i i 1?! KK(D; A A 1 ) j?j 1?! KK(D; A? A 1 )! KK(C; D) i +i 1? KK(A A 1 ; D) j?j 1? KK(A? A 1 ; D) 6 Applications One of the incentives of this work was to prove the following Corollary 6.1 The C -algebra of a nite family of semi-circular elements is K-equivalent to C; which trivially implies the result of [21]. Proof The C -algebra of n semi-circular elements can be represented by? n i=1;red C([; 1]). It is thus K-equivalent to?n i=1;f ullc([; 1]) which is contractible. Remark 6.2 The element of KK(? n i=1;redc([; 1]); C) given by the restriction to the semi-circular elements of the K-bimodule t red of Cuntz (see [3]) for the free group F n realizes the K-equivalence. Proposition 6.3 Let G be the group of permutations of a set of n elements, and take n identical unital nuclear separable C -algebras endowed 18
with the same state then G acts on the reduced free product and the canonical map of the full free product into the reduced one is an equivalence in G-equivariant K-theory. Proof All the bimodules used in sections 2 and 3 were G-equivariant. If A is a unital separable nuclear C -algebra endowed Corollary 6.4 with a state (such that the associated GNS representation is faithfull) then for any n (A?n ) G is K-equivalent to (? n i=1;red (A; ')) G: Remark 6.5 Using the results of section 3 and 5, one can reprove the result of Cuntz that a free product of K-amenable discrete groups is again K-amenable (recall that K-amenable implies K-nuclear). In [5], we are able to prove a generalization of theorem Remark 6.6 4.1 to continuous elds of nuclear C -algebras and reduced free products amalgamated over C(X): References [1] D. Avitzour Free products of C -algebras Trans. Amer. Math. Soc 271 (1982) 423-465 [2] B. Blackadar K-theory of operator algebras, MSRI Publications n 5 (1986) [3] J. Cuntz K-theoretic amenability for discrete groups Crelles J. 344 (1983) 18-195 [4] J. Cuntz The K-groups for free products of C -algebras, Proceedings of Symposia in Pure Mathematics, Vol 38, Part 1, A.M.S. (1982) [5] E. Germain KK-theory of reduced free product C -algebras with amalgamation over C(X) Ph'D Dissertation University of California at Berkeley 1994 [6] P. Julg A. Valette K-theoretic amenability for SL 2 (Q p ) and the action on the associated tree J. of Functional Analysis 58 (1984) 194-215 [7] G. Kasparov Hilbert C -modules: Theorems of Stinespring and Voiculescu, J. of Operator Theory 4 (198) 133-15 [8] G. Kasparov Operator K-functor and extension of C -algebras, Math. USSR Izvestia 16 (1981) 513-572 19
[9] E. Lance K-theory for certain group C -algebras Acta Math. 151 (1983) 29-23 [1] K. McClanahan K-theory for certain reduced free products of C - algebras Preprint University of Mississippi (1992) [11] K. McClanahan KK-group of crossed products by grouplike sets acting on trees Preprint University of Mississippi (1993) [12] J. Mingo On the contractibility of the unitary group of the Hilbert space over a C -algebra Integral Eq. Operator Th. 5 (1982) 888-891 [13] M. Pimsner KK-group of crossed products by groups acting on trees Inventionnes Math. 86 (1986) 63-634 [14] M. Pimsner, D. Voiculescu K-groups of reduced crossed products by free groups Journal of Operator Theory 8 (1982) 131-156 [15] J. Rosenberg C. Schochet The Kunneth theorem and the universal coecients theorem for the Kasparov's generalized K-functor Duke Math J. 55 (1987) 431-474 [16] G. Skandalis Une Notion de nuclearite en K-theorie (d'apres J. Cuntz), K-theory 1 (1988) 549-573 [17] G. Skandalis Kasparov's bivariant K-theory and Applications Expositiones Mathematicae vol. 9 n 3 (1991) [18] E.H. Spanier Algebraic Topology Springer 1966 [19] D. Voiculescu A non-commutative Weyl-Von Neumann theorem Revue Roumaine Math. Pures Appl. 21 (1976) 97-113 [2] D. Voiculescu Symetries of some reduced free product C - algebras, Operator Algebras and Their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics, Vol 1132, Springer (1985) [21] D. Voiculescu The K-groups of the C -algebra of a semicircular family K-theory 7 (1993) 5-7 2