Tensor products of C(X)-algebras over C(X)

Transcription

1 Tensor products of C(X)-algebras over C(X) Etienne Blanchard To cite this version: Etienne Blanchard. Tensor products of C(X)-algebras over C(X). Asterisque, Société Mathéatique de France, 1995, 232, pp <hal v2> HAL Id: hal Subitted on 3 Jan 2014 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Tensor products of C(X)-algebras over C(X) Etienne Blanchard Abstract Given a Hausdorff copact space X, we study the C -(sei)-nors on the algebraic tensor product A C(X) B of two C(X)-algebras A and B over C(X). In particular, if one of the two C(X)-algebras defines a continuous field of C - algebras over X, there exist inial and axial C -nors on A C(X) B but there does not exist any C -nor on A C(X) B in general. AMS classification: 46L05, 46M05. 0 Introduction Tensor products of C -algebras have been extensively studied over the last decades (see references in [12]). One of the ain results was obtained by M. Takesaki in [15] where he proved that the spatial tensor product A in B of two C -algebras A and B always defines the inial C -nor on the algebraic tensor product A B of A and B over the coplex field C. More recently, G.G. Kasparov constructed in [10] a tensor product over C(X) for C(X)-algebras. The author was also led to introduce in [3] several notions of tensor products over C(X) for C(X)-algebras and to study the links between those objects. Notice that E. Kirchberg and S. Wasserann have proved in [11] that the subcategory of continuous fields over a Hausdorff copact space is not closed under such tensor products over C(X) and therefore, in order to study tensor products over C(X) of continuous fields, it is natural to work in the C(X)-algebras fraework. Let us introduce the following definition: DEFINITION 0.1 Given two C(X)-algebras A and B, we denote by I(A, B) the involutive ideal of the algebraic tensor product A B generated by the eleents (f a) b a (fb), where f C(X), a A and b B. Our ai in the present article is to study the C -nors on the algebraic tensor product (A B)/I(A, B) of two C(X)-algebras A and B over C(X) and to see how one can enlarge the results of Takesaki to this fraework. We first define an ideal J (A,B) A B which contains I(A,B) such that every C -sei-nor on A B which is zero on I(A,B) is also zero on J (A,B) and we prove that there always exist a inial C -nor and a axial C -nor M on the quotient (A B)/J (A, B). 1

3 We then study the following question of G.A. Elliott ([5]): when do the two ideals I(A,B) and J (A,B) coincide? The author would like to express his gratitude to C. Anantharaan-Delaroche and G. Skandalis for helpful coents. He is also very indebted to S. Wasserann for sending hi a preliinary version of [11] and to J. Cuntz who invited hi to the Matheatical Institute of Heidelberg. 1 Preliinaries We briefly recall here the basic properties of C(X)-algebras. Let X be a Hausdorff copact space and C(X) be the C -algebra of continuous functions on X. For x X, define the orphis e x : C(X) C of evaluation at x and denote by C x (X) the kernel of this ap. DEFINITION 1.1 ([10]) A C(X)-algebra is a C -algebra A endowed with a unital orphis fro C(X) in the center of the ultiplier algebra M(A) of A. We associate to such an algebra the unital C(X)-algebra A generated by A and u[c(x)] in M[A C(X)] where u(g)(a f) = ga gf for a A and f,g C(X). For x X, denote by A x the quotient of A by the closed ideal C x (X)A and by a x the iage of a A in the fibre A x. Then, as a x = inf{ [1 f + f(x)]a,f C(X)}, the ap x a x is upper sei-continuous for all a A ([14]). Note that the ap A A x is a onoorphis since if a A, there is a pure state φ on A such that φ(a a) = a 2. As the restriction of φ to C(X) M(A) is a character, there exists x X such that φ factors through A x and so φ(a a) = a x 2. Let S(A) be the set of states on A endowed with the weak topology and let S X (A) be the subset of states ϕ whose restriction to C(X) M(A) is a character, i-e such that there exists an x X (denoted x = p(ϕ)) verifying ϕ(f) = f(x) for all f C(X). Then the previous paragraph iplies that the set of pure states P(A) on A is included in S X (A). Let us introduce the following notation: if E is a Hilbert A-odule where A is a C - algebra, we will denote by L A (E) or siply L(E) the set of bounded A-linear operators on E which adit an adjoint ([9]). DEFINITION 1.2 ([3]) Let A be a C(X)-algebra. A C(X)-representation of A in the Hilbert C(X)-odule E is a orphis π : A L(E) which is C(X)-linear, i.e. such that for every x X, the representation π x = π e x in the Hilbert space E x = E ex C factors through a representation of A x. Furtherore, if π x is a faithful representation of A x for every x X, π is said to be a field of faithful representations of A. A continuous field of states on A is a C(X)-linear ap ϕ : A C(X) such that for any x X, the ap ϕ x = e x ϕ defines a state on A x. 2

4 If π is a C(X)-representation of the C(X)-algebra A, the ap x π x (a) is lower sei-continuous since ξ, π(a)η C(X) for every ξ, η E. Therefore, if A adits a field of faithful representations π, the ap x a x = π x (a) is continuous for every a A, which eans that A is a continuous field of C -algebras over X ([4]). The converse is also true ([3] théorèe 3.3): given a separable C(X)-algebra A, the following assertions are equivalent: 1. A is a continuous field of C -algebras over X, 2. the ap p : S X (A) X is open, 3. A adits a field of faithful representations. 2 C -nors on (A B)/J (A,B) DEFINITION 2.1 Given two C(X)-algebras A and B, we define the involutive ideal J (A,B) of the algebraic tensor product A B of eleents α A B such that α x = 0 in A x B x for every x X. By construction, the ideal I(A,B) is included in J (A,B). PROPOSITION 2.2 Assue that β is a C -sei-nor on the algebraic tensor product A B of two C(X)-algebras A and B. If β is zero on the ideal I(A,B), then α β = 0 for all α J (A,B). Proof: Let D β be the Hausdorff copletion of A B for β. By construction, D β is a quotient of A ax B. Furtherore, if C is the ideal of C(X X) of functions which are zero on the diagonal, the iage of C in M(D β ) is zero. As a consequence, the ap fro A ax B onto D β factors through the quotient A M C(X) B of A ax B by C (A ax B). But an easy diagra-chasing arguent shows that (A M C(X) B) x = A x ax B x for every x X ([3] corollaire 3.17) and therefore the iage of J (A,B) A ax B in A M C(X) B is zero. 2.1 The axial C -nor DEFINITION 2.3 Given two C(X)-algebras A 1 and A 2, we denote by M the C - sei-nor on A 1 A 2 defined for α A 1 A 2 by where σ x i is the ap A i (A i ) x. α M = sup{ (σ x 1 ax σ x 2)(α),x X} As M is zero on the ideal J (A 1,A 2 ), if we identify M with the C -sei-nor induced on (A 1 A 2 )/J (A 1,A 2 ), we get: 3

5 PROPOSITION 2.4 The sei-nor M is the axial C -nor on the quotient (A 1 A 2 )/J (A 1,A 2 ). Proof: By construction, M defines a C -nor on (A 1 A 2 )/J (A 1,A 2 ). Moreover, as the quotient A 1 M C(X) A 2 of A 1 ax A 2 by C (A 1 ax A 2 ) aps injectively in M (A 1 C(X) A 2 ) x = ( (A 1 ) x ax (A 2 ) x ), x X x X the nor of A 1 M C(X) A 2 coincides on the dense subalgebra (A 1 A 2 )/J (A 1,A 2 ) with M. But we saw in proposition 2.2 that if β is a C -nor on the algebra (A B)/J (A,B), the copletion of (A B)/J (A, B) for β is a quotient of A M C(X) B. 2.2 The inial C -nor DEFINITION 2.5 Given two C(X)-algebras A 1 and A 2, we define the sei-nor on A 1 A 2 by the forula α = sup{ (σ x 1 in σ x 2)(α),x X} where σ x i is the ap A i (A i ) x and we denote by A 1 C(X) A 2 the Hausdorff copletion of A 1 A 2 for that sei-nor. Reark: In general, the canonical ap (A 1 C(X) A 2 ) x (A 1 ) x in (A 2 ) x is not a onoorphis ([11]). By construction, induces a C -nor on (A 1 A 2 )/J (A 1,A 2 ). We are going to prove that this C -nor defines the inial C -nor on the involutive algebra (A 1 A 2 )/J (A 1,A 2 ). Let us introduce soe notation. Given two unital C(X)-algebras A 1 and A 2, let P(A i ) S X (A i ) denote the set of pure states on A i and let P(A 1 ) X P(A 2 ) denote the closed subset of P(A 1 ) P(A 2 ) of couples (ω 1,ω 2 ) such that p(ω 1 ) = p(ω 2 ), where p : P(A i ) X is the restriction to P(A i ) of the ap p : S X (A i ) X defined in section 1. LEMMA 2.6 Assue that β is a C -sei-nor on the algebraic tensor product A 1 A 2 of two unital C(X)-algebras A 1 and A 2 which is zero on the ideal J (A 1,A 2 ) and define the closed subset S β P(A 1 ) X P(A 2 ) of couples (ω 1,ω 2 ) such that (ω 1 ω 2 )(α) α β for all α A 1 A 2. If S β P(A 1 ) X P(A 2 ), there exist self-adjoint eleents a i A i such that a 1 a 2 J (A 1,A 2 ) but (ω 1 ω 2 )(a 1 a 2 ) = 0 for all couples (ω 1,ω 2 ) S β. Proof: Define for i = 1, 2 the adjoint action ad of the unitary group U(A i ) of A i on the pure states space P(A i ) by the forula 4

6 [(ad u )ω](a) = ω(u au). Then S β is invariant under the product action ad ad of U(A 1 ) U(A 2 ) and we can therefore find non epty open subsets U i P(A i ) which are invariant under the action of U(A i ) such that (U 1 U 2 ) S β =. Now, if K i is the copleent of U i in P(A i ), the set K i = {a A i ω(a) = 0 for all ω K i } is a non epty ideal of A i and furtherore, if ω P(A i ) is zero on Ki, then ω belongs to K i ([8] lea 8,[15]). As a consequence, if (ϕ 1,ϕ 2 ) is a point of U 1 X U 2, there exist non zero self-adjoint eleents a i Ki such that ϕ i (a i ) = 1. If x = p(ϕ i ), this iplies in particular that (a 1 ) x (a 2 ) x 0, and hence a 1 a 2 J (A 1,A 2 ). LEMMA 2.7 ([15] theore 1) Let A 1 and A 2 be two unital C(X)-algebras. If the algebra A 1 is an abelian algebra, there exists only one C -nor on the quotient (A 1 A 2 )/J (A 1,A 2 ). Proof: Let β be a C -sei-nor on A 1 A 2 such that for all α A 1 A 2, α β = 0 if and only if α J (A 1,A 2 ). If ρ P(A β ) is a pure state on the Hausdorff copletion A β of A 1 A 2 for the sei-nor β, then for every a 1 a 2 A 1 A 2, ρ(a 1 a 2 ) = ρ(a 1 1)ρ(1 a 2 ) since A 1 1 is included in the center of M(A β ). Moreover, if we define the states ω 1 and ω 2 by the forulas ω 1 (a 1 ) = ρ(a 1 1) and ω 2 (a 2 ) = ρ(1 a 2 ), then ω 2 is pure since ρ is pure, and (ω 1,ω 2 ) P(A 1 ) X P(A 2 ). It follows that P(A β ) is isoorphic to S β. In particular, if a 1 a 2 A 1 A 2 verifies (ω 1 ω 2 )(a 1 a 2 ) = 0 for all couples (ω 1,ω 2 ) S β, the eleent a 1 a 2 is zero in A β and therefore belongs to the ideal J (A 1, A 2 ). Accordingly, the previous lea iplies that P(A 1 ) X P(A 2 ) = S β = P(A β ). As a consequence, we get for every α A 1 A 2 α 2 β = sup{ρ(α α),ρ P(A β )} = sup{(ω 1 ω 2 )(α α), (ω 1,ω 2 ) P(A 1 ) X P(A 2 )} But that last expression does not depend on β, and hence the unicity. PROPOSITION 2.8 ([15] theore 2) Let A 1 and A 2 be two unital C(X)-algebras. If β is a C -sei-nor on A 1 A 2 whose kernel is J (A 1,A 2 ), then α A 1 A 2, α β α. Proof: If we show that S β = P(A 1 ) X P(A 2 ), then for every ρ S β and every α in A 1 A 2, we have ρ(s α αs) ρ(s s) α 2 β for all s A 1 A 2. Therefore 5

7 α 2 = sup { (σ x 1 in σ x 2)(α) 2,x X } = sup { (ω 1 ω 2 )(s α αs), (ω (ω 1 ω 2 )(s 1, ω 2 ) P(A 1 ) X P(A 2 ) and s) s A 1 A 2 such that (ω 1 ω 2 )(s s) 0 } α 2 β. Suppose that S β P(A 1 ) X P(A 2 ). Then there exist thanks to lea 2.6 self-adjoint eleents a i A i and a point x X such that (a 1 ) x (a 2 ) x 0 but (ω 1 ω 2 )(a 1 a 2 ) = 0 for all couples (ω 1,ω 2 ) S β. Let B be the unital abelian C(X)-algebra generated by C(X) and a 1 in A 1. The preceding lea iplies that B C(X) A 2 aps injectively into the Hausdorff copletion A β of A 1 A 2 for β. Consider pure states ρ P(B x ) and ω 2 P( (A 2 ) x ) such that ρ(a 1 ) 0 and ω 2 (a 2 ) 0 and extend the pure state ρ ω 2 on B C(X) A 2 to a pure state ω on A β. If we set ω 1 (a) = ω(a 1) for a A 1, then ω 1 is pure and ω(α) = (ω 1 ω 2 )(α) for all α A 1 A 2 since ω and ω 2 are pure ([15] lea 4). As a consequence, (ω 1, ω 2 ) S β, which is absurd since (ω 1 ω 2 )(a 1 a 2 ) = ρ(a 1 )ω 2 (a 2 ) 0. PROPOSITION 2.9 Given two C(X)-algebras A 1 and A 2, the sei-nor defines the inial C -nor on the involutive algebra (A 1 A 2 )/J (A 1,A 2 ). Proof: Let β be a C -nor on (A 1 A 2 )/J (A 1,A 2 ). Thanks to the previous proposition, all we need to prove is that one can extend β to a C -nor on (A 1 A 2 )/J (A 1, A 2 ), where A 1 and A 2 are the unital C(X)-algebras associated to the C(X)-algebras A 1 and A 2 (definition 1.1). Consider the Hausdorff copletion D β of (A 1 A 2 )/J (A 1,A 2 ) and denote by π i the canonical representation of A i in M(D β ) for i = 1, 2. Let us define the representation π i of A i in M(D β A 1 A 2 C(X)) by the following forulas: π 1 (b 1 + u(f))(α a 1 a 2 g) = (π 1 (b 1 ) + g)α (b 1 + f)a 1 fa 2 fg π 2 (b 2 + u(f))(α a 1 a 2 g) = (π 2 (b 2 ) + g)α fa 1 (b 2 + f)a 2 fg For i = 1, 2, let ε i : A i C(X) be the ap defined by ε i [a + u(f)] = f for a A i and f C(X). Then using the aps (ε 1 ε 2 ), (ε 1 id) and (id ε 2 ), one proves easily that if α A 1 A 2, ( π 1 π 2 )(α) = 0 if and only if α belongs to J (A 1, A 2 ). Therefore, the nor of M(D β A 1 A 2 C(X)) restricted to the subalgebra (A 1 A 2 )/J (A 1, A 2 ) extends β. Reark: As the C(X)-algebra A is nuclear if and only if every fibre A x is nuclear ([12]), A M C(X) B A C(X) B for every C(X)-algebra B if and only if A is nuclear. 6

8 3 When does the equality I(A, B) =J (A, B) hold? Given two C(X)-algebras A and B, Giordano and Mingo have studied in [6] the case where the algebra C(X) is a von Neuann algebra: their theore 3.1 and lea 1.5 of [10] iply that in that case, we always have the equality I(A,B) = J (A,B). Our purpose in this section is to find sufficient conditions on the C(X)-algebras A and B in order to ensure this equality and to present a counter-exaple in the general case. PROPOSITION 3.1 Let X be a second countable Hausdorff copact space and let A and B be two C(X)-algebras. If A is a continuous field of C -algebras over X, then I(A,B) = J (A,B). Proof: Let us prove by induction on the non negative integer n that if s = a i b i J (A,B), then s belongs to the ideal I(A,B). 1 i n If n = 0, there is nothing to prove. Consider therefore an integer n > 0 and suppose the result has been proved for any p < n. Fix an eleent s = a i b i J (A,B) and define the continuous positive 1 i n function h C(X) by the forula h(x) 10 = (a k ) x 2. The eleent a k = h 4 a k is then well defined in A for every k since a ka k h 10. Consequently, the function f k (x) = (a k) x is continuous. For 1 k n, let D k denote the separable C(X)-algebra generated by 1 and the a k a j, 1 j n, in the unital C(X)-algebra A associated to A (definition 1.1). Then D k is a unital continuous field of C -algebras over X (see for instance [3] proposition 3.2). Consider the open subset S k = {ψ S X (D k )/ψ[a k a k] > ψ(f 2 k/2)}. If we apply lea 3.6 b) of [3] to the restriction of p : S X (D k ) X to S k, we ay construct a continuous field of states ω k on D k such that ω k [a k a k] f 2 k/2. Now, if we set s = i a i b i, as (a k 1)s belongs to J (D k,b), (ω k id)[(a k 1)s ] = ω k [a k a k]b k + ω k [a k a j]b j = 0. Noticing that f 3 k is in the ideal of C(X) generated by ω k [a k a k], we get that f 3 kb k belongs to the C(X)-odule generated by the b j, j k, and thanks to the induction hypothesis, it follows that (f 3 k 1)s I(A,B) for each k. But h 2 = k f 2 k and so h 4 n k f 4 k is in the ideal of C(X) generated by the f 3 k, which iplies s = (h 4 1)s I(A,B). Rearks: 1. As a atter of fact, it is not necessary to assue that the space X satisfies the second axio of countability thanks to the following lea of [11]: if P(a) C(X) 7 j k

9 denotes the ap x a x for a A, there exists a separable C -subalgebra C(Y ) of C(X) with sae unit such that if D k is the separable unital C -algebra generated by C(Y ).1 A and the a k a j, 1 j n, then P(D k ) = C(Y ). Furtherore, if Φ : X Y is the transpose of the inclusion ap C(Y ) C(X) restricted to pure states, the ap D/(C Φ(x) (Y )D) A x is a onoorphis for every x X since A is continuous. Consider now a continuous field of states ω k : D k C(Y ) on the continuous field D k over Y ; if c j d j D k B is zero in A x B x for x X, then ω k (c j )(x)d j x = 0 in B x, which enables us to conclude as in the separable case. 2. J. Mingo has drawn the author s attention to the following result of Gli ([7] lea 10): if C(X) is a von Neuann algebra and A is a C(X)-algebras, then P(a) 2 = in{z C(X) +, z a a} is continuous for every a A. Therefore we always have in that case the equality I(A, B) = J (A,B) thanks to the previous reark. COROLLARY 3.2 Let A and B be two C(X)-algebras and assue that there exists a finite subset F = {x 1,,x p } X such that for all a A, the function x a x is continuous on X\F. Then the ideals I(A, B) and J (A, B) coincide. Proof: Fix an eleent α = 1 i n a i b i A B which belongs to J (A,B). In particular, we have i(a i ) x (b i ) x = 0 in A x B x for each x F. As a consequence, thanks to theore III of [13], we ay find coplex atrices (λ i,j) i,j M n (C) for all 1 p such that, if we define the eleents c k A and d k B by the forulas c k = i λ i,ka i and d k = b k j λ k,jb j, we have (c k ) x = 0 and (d k ) x = 0 for all k and all. Consider now a partition {f l } 1 l p of 1 C(X) such that for all 1 l, p, f l (x ) = δ l, where δ is the Kronecker sybol and define for all 1 k n the eleents c k = f c k and d k = f d k. Thus, α = ( i a i d i ) + ( i,j, λ i,ja i f b j ) = ( i a i d i ) + ( j c j b j ) + ( i,j, λ i,j(a i f b j f a i b j ) ) and there exists therefore an eleent β I(A, B) such that α β adits a finite decoposition i a i b i with a i C 0 (X\F)A and b i C 0 (X\F)B. But C 0 (X\F)A is a continuous field. Accordingly, proposition 3.1 iplies that α β I(C 0 (X\F)A,C 0 (X\F)B) I(A,B). Reark: If Ñ = N { } is the Alexandroff copactification of N and if A and B are two C(Ñ)-algebras, the corollary 3.2 iplies the equality I(A,B) = J (A,B). Let us now introduce a counter-exaple in the general case. Consider a dense countable subset X = {a n } n N of the interval [0, 1]. The C - algebra C 0 (N) of sequences with values in C vanishing at infinity is then endowed with the C([0, 1])-algebra structure defined by: 8

10 f C([0, 1]), α = (α n ) C 0 (N), (f.α) n = f(a n )α n for n N. If we call A this C([0, 1])-algebra, then A x = 0 for all x X. Indeed, assue that x X and take α A. If ε > 0, there exists N N such that α n < ε for all n N. Consider a continuous function f C([0, 1]) such that 0 f 1, f(x) = 0 and f(a i ) = 1 for every 1 i N; we then have: α x (1 f)α < ε. Let Y = {b n } n N be another dense countable subset of [0, 1] and denote by B the associated C([0, 1])-algebra whose underlying algebra is C 0 (N). Then, if X Y =, the previous reark iplies that for every x [0, 1], A x B x = 0 and hence, J (A,B) = A B. What we therefore need to prove is that the ideal I(A, B) is strictly included in A B. Let us fix two sequences α A and β B whose ters are all non zero and suppose that α β adits a decoposition 1 i k[(f i α i ) β i α i (f i β i )] in A B. Then for every n, N, α n β = αnβ i [f i i (a n ) f i (b )]. 1 i k Now, if we set φ i (a n ) = α i n/α n and ψ i (b n ) = β i n/β n for 1 i k and n N, this equality eans that for all (x, y) X Y, 1 = 1 i kψ i (x)ϕ i (y)[f i (x) f i (y)]. But this is ipossible because of the following proposition: PROPOSITION 3.3 Let X and Y be two dense subsets of the interval [0, 1] and let n be a non negative integer. Given continuous functions f i on [0, 1] and nuerical functions ψ i : X C and ϕ i : Y C for 1 i n, if there exists a constant c C such that then c = 0. (x,y) X Y, 1 i n ψ i (x)ϕ i (y)[f i (x) f i (y)] = c Proof: We shall prove the proposition by induction on n. If n = 0, the result is trivial. Take therefore n > 0 and assue that the proposition is true for any k < n. Suppose then that the subsets X and Y of [0, 1], the functions f i, ψ i and ϕ i, 1 i n, satisfy the hypothesis of the proposition for the constant c. For x X, let p(x) n be the diension of the vector space generated in C n by the ( ϕ i (y)[f i (x) f i (y)] ) 1 i n, y Y. If p(x) < n, there exists a subset F(x) {1,...,n} of cardinal p(x) such that for every j F(x): 9

11 ϕ j (y)[f j (x) f j (y)] = i F(x) λ j i(x)ϕ i (y)[f i (x) f i (y)] for all y Y, where the λ j i(x) C are given by the Craer forulas. As a consequence, ( i F(x) ψ i (x) + ) j F(x)[λ j i(x)ψ j (x)] ϕ i (y)[f i (x) f j (y)] = c. Now, if p(x) < n for every x X, there exists a subset F {1,...,n} of cardinal p < n such that the interior of the closure of the set of those x for which F(x) = F is not epty and contains therefore a closed interval hoeoorphic to [0, 1]. The induction hypothesis for k = p iplies that c = 0. Assue on the other hand that x 0 X verifies p(x 0 ) = n. We ay then find y 1, y n in Y such that if we set a i,j (x) = ϕ i (y j )[f i (x) f i (y j )], the atrix (a i,j (x 0 )) is invertible. There exists therefore a closed connected neighborhood I of x 0 on which the atrix (a i,j (x)) reains invertible. But for each 1 j n, a i,j (x)ψ i (x) = c and therefore the ψ i (x) extend by the i Craer forulas to continuous functions on the closed interval I. For y Y I, let q(y) denote the diension of the vector space generated in C n by the ( ψ i (x)[f i (x) f i (y)] ) 1 i n, x X I. If q(y) < n for every y, then the induction hypothesis iplies c = 0. But if there exists y 0 such that q(y 0 ) = n, we ay find an interval J I hoeoorphic to [0, 1] on which the ϕ i extend to continuous functions; evaluating the starting forula at a point (x,x) J J, we get c = 0. 4 The associativity Given three C(X)-algebras A 1, A 2 and A 3, we deduce fro [3] corollaire 3.17: [(A 1 M C(X) A 2 ) M C(X) A 3 ] x =(A 1 M C(X) A 2 ) x ax (A 3 ) x =(A 1 ) x ax (A 2 ) x ax (A 3 ) x, which iplies the associativity of the tensor product M C(X) over C(X). On the contrary, the inial tensor product C(X) over C(X) is not in general associative. Indeed, Kirchberg and Wasserann have shown in [11] that if Ñ = N { } is the Alexandroff copactification of N, there exist separable continuous fields A and B such that (A C(Ñ) B) A in B. If we now endow the C -algebra D = C with the C(Ñ)-algebra structure defined by f.a = f( )a, then for all C(Ñ)-algebra D, we have 10

12 [D C(Ñ) D ] n = { 0 if n is finite, (D ) if n =. Therefore, [(A C(Ñ) B) C(Ñ) D] (A C(Ñ) B) whereas [A C(Ñ) (B C(Ñ) D)] is isoorphic to A in B. However, in the case of (separable) continuous fields, we can deduce the associativity of C(X) fro the following proposition: PROPOSITION 4.1 Let A and B be two C(X)-algebras. Assue π is a field of faithful representations of A in the Hilbert C(X)-odule E, then the orphis a b π(a) b induces a faithful C(X)-linear representation of A C(X) B in the Hilbert B-odule E C(X) B. Proof: Notice that for all x X, we have (E C(X) B) B B x = E x B x. Now, as B aps injectively in B d = x X B x, L B (E C(X) B) aps injectively in x X L Bx (E x B x ) L Bd (E C(X) B B B d ) and therefore if α A B, we have (π id)(α) = sup (π x id)(α x ) = α. x X Accordingly, if for 1 i 3, A i is a separable continuous field of C -algebras over X which adits a field of faithful representations in the C(X)-odule E i, the C(X)-representations of (A 1 C(X) A 2 ) C(X) A 3 and A 1 C(X) (A 2 C(X) A 3 ) in the Hilbert C(X)-odule (E 1 C(X) E 2 ) C(X) E 3 = E 1 C(X) (E 2 C(X) E 3 ) are faithful, and hence the aps A 1 in A 2 in A 3 (A 1 C(X) A 2 ) C(X) A 3 and A 1 in A 2 in A 3 A 1 C(X) (A 2 C(X) A 3 ) have the sae kernel. References [1] B. BLACKADAR K-theory for Operator Algebras, M.S.R.I. Publications 5, Springer Verlag, New York (1986). [2] E. BLANCHARD Représentations de chaps de C -algèbres, C.R.Acad.Sc. Paris 314 (1992), [3] E. BLANCHARD Déforations de C -algèbres de Hopf, thèse de Doctorat (Paris VII, nov. 1993). [4] J. DIXMIER Les C -algèbres et leurs représentations, Gauthiers-Villars Paris (1969). [5] G.A. ELLIOTT Finite Projections in Tensor Products von Neuann Algebras, Trans. Aer. Math. Soc. 212 (1975), [6] T. GIORDANO and J.A. MINGO Tensor products of C -algebras over abelian subalgebras, preprint, Queen s University, April

13 [7] J. GLIMM A Stone-Weierstrass theore for C -algebras, Ann. of Math. 72 (1960), [8] J. GLIMM Type I C -algebras, Ann. Math. 73 (1961), [9] G.G. KASPAROV Hilbert C -odules: theores of Stinespring and Voiculescu, J. Operator Theory 4 (1980), [10] G.G. KASPAROV Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), [11] E. KIRCHBERG and S. WASSERMANN Operations on continuous bundles of C -algebras, preprint (preliinary version), Septeber [12] C. LANCE Tensor product and nuclear C -algebras, Operator Algebras and Applications, Proc. Syposia Pure Math 38 (1982) Part I, [13] F.J. MURRAY and J. von NEUMANN On rings of Operators, Ann. of Math. 37 (1936), [14] M.A. RIEFFEL Continuous fields of C -algebras coing fro group cocycles and actions, Math. Ann. 283 (1989), [15] M. TAKESAKI On the cross nor of the direct product of C -algebras, Tôhoku Math J. 16 (1964), Etienne Blanchard, Matheatisches Institut, Universität Heidelberg I Neuenheier Feld 288, D Heidelberg e-ail: blanchar@athi.uni-heidelberg.de 12