C M OMPOSITIO ATHEMATICA CLAIRE ANANTHARAMAN-DELAROCHE OnrelativeamenabilityforvonNeumannalgebras CompositioMathematica,tome 74,no3(1990),p. 333-352 <http://www.numdam.org/item?id=CM_1990__74_3_333_0> ©FoundationCompositioMathematica,1990,tousdroitsréservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 74:3 33333-352, 1990. (Ç) 1990 Kluwer Academic Publishers. Printed in the Netherlands. On relative amenability for von Neumann algebras CLAIRE ANANTHARAMAN-DELAROCHE Universite d’Orléans, U.F.R. Faculté des Sciences, Département de Mathématiques et Informatique, B.P. 6759, F-45067 Orleans CEDEX 2 Received 9 November 1988; accepted 19 September 1989 Introduction The concept of correspondence between two von Neumann algebras has been introduced by A. Connes ([8], [9]) as a very useful tool for the study of type II1 factors. Recently, S. Popa has systematically developed this point of view to get some new insight in the domain [21]. Among many interesting results and remarks, he discussed Connes’ classical work on the injective II1 factor in the framework of correspondences, and he defined and studied a natural notion of amenability for a finite von Neumann algebra M relative to a von Neumann subalgebra N. When the Jones’ index [M : N] is finite or when M is injective the inclusion N c M is amenable, but this situation occurs in many other examples. For instance, if M is the crossed product of a finite von Neumann algebra N by an action of a discrete group G preserving a faithful finite normal trace of N, then N c M is amenable if and only if G is an amenable group ([21], Th. 3.2.4). In [28], Zimmer considered a notion of amenable action in ergodic theory, which was extended in [1] to actions on arbitrary von Neumann algebras. We say that the G-action a on N is amenable if there exists an equivariant norm one projection from L~(G) ~ N onto N, the G-action on L~(G) ~ N being the tensor product of the action by left translation on L°°(G) and the action a on N. When there exists a G-invariant state on the centre Z(N) of N, the amenability of the action is equivalent to the amenability of the group ([1], Prop. 3.6). Otherwise, it is easy to construct amenable actions of non amenable groups. Since Popa’s notion of amenable inclusion makes sense for arbitrary von Neumann algebras, he asked ([21], 3.4.2) whether the amenability of the G-action oc was equivalent to the amenability of the inclusion N c M = N 03B1 G in the case of a discrete group G acting on any von Neumann algebra N. In this paper we give a positive answer to this question (Prop. 3.4). As far as we are concerned with non finite von Neumann algebras M and N, it seems more convenient to consider a correspondence between M and N as a self-dual right Hilbert N-module on which M acts to the left, since it avoids the choice of auxiliary weights. This point of view has been already systematically used in [4] for the general study of the index of conditional expectations. In the 334 first section we recall the needed background on correspondences and Hilbert modules. In particular, to any inclusion N c M is associated a correspondence YN (see 1.8) between M and M which gives informations about the embedding N c M. Popa has defined the inclusion to be amenable if the identity correspondence of M is weakly contained in YN. In Section 2 we consider an action a of a discrete group G on a von Neumann algebra N, and we denote by M the crossed product N xa G. The classical notions of positive type functions and group representations can respectively be extended in this context of dynamical systems to notions of positive type functions on G with respect to (N, G, oc) and of cocycles (2.4 and 2.1). These two concepts are closely related, as in the usual case. For each cocycle T relative to (Z(N), G, a) we associate in a natural way a correspondence X between M and M (2.6). A positive type function corresponding to T gives rise to a normal completely positive map from M to M, which is a coefficient of the correspondence X (2.8). The positive type functions relative to (Z(N), G, a), having finite supports, yield coefficients of the correspondence YN associated to the inclusion N c M, and the constant positive type function equal to the unit of Z(N) gives the identity automorphism of M, which is, of course, a coefficient of the identity correspondence of M. We proved in [3] that the G-action a on N is amenable if and only if this constant function is the limit, for the topology of the Q-weak pointwise convergence, of a net of positive type functions relative to (Z(N), G, a) with finite supports. Using this fact, we show in Section 3 the equivalence between the amenability of the action and the amenability of the inclusion N c M. 1. Preliminaries on correspondences We recall here some facts on correspondences and Hilbert modules, mostly coming from [8], [9], [4], [21], [20], [22], [23], [24], [17], where the reader will find more details. For simplicity, in this paper we shall only consider a-finite von Neumann algebras. Let M and N be two von Neumann algebras. 1.1. A correspondence between M and N is a Hilbert space H with a pair of commuting normal representations 03C0M and 1CNO of M and N° (the opposite of N) respectively [8]. Usually the triple (H, xM, nNO) will be denoted by H, and for x E M, y E N and h E H, we shall write xhy instead of 03C0M(x)03C0N0(y0)h. Note that H gives rise to a representation of the binormal tensor product M ~bin N° (see [ 11 ] for the definition of bin). Two correspondences H and H’ are equivalent if they are (unitarily) equivalent when considered as representations of M ~binN0. We denote by Corr(M, N) the set of equivalence classes of correspondences between M and N, and we shall use the same notation H for a correspondence 335 and its class. We shall write Corr(M) for Corr(M, M). The standard form [ 13] of M yields an element L2(M) of Corr(M) called the identity correspondence of M. We shall sometimes write L2(M, cp) instead of L2(M), with a fixed faithful normal positive form cp on M. 1.2. Let us recall now another useful equivalent way of defining correspondences, which has been developed in [4]. Let X be a self-dual (right) Hilbert N-module (see [20]). We denote by ,&#x3E; (or ,,) N in case of ambiguity) the N-valued inner product, and we suppose that it is conjugate linear in the first variable. The von Neumann algebra of all N-linear continuous operators from X to X will be denoted by N(X) (or (X) when N = C). Following ([4], Def. 2.1), by a M-N correspondence we mean a pair (X, rc) where X is as above, and n is a unital normal homomorphism from M into N(X). More briefly, such a correspondence will be denoted by X, and we shall often write x03BE instead of 03C0(x)03BE for x E M and 03BE E X. Let us remark that M-N correspondences are what Rieffel has called normal N-rigged M-modules in ([23], Def. 5.1). Two M-N correspondences X and X’ are said to be equivalent if there exists a M-N linear isomorphism from X onto X’ preserving the scalar products. 1.3. Let X be a self-dual Hilbert N-module. We call s-topology the topology defined on X by the family of semi-norms q~, where 9 is any normal positive form on N and We say that a vector 03BE in a M-N correspondence X is cyclic if the set M03BEN {x03BEy, x ~ M, y c- NI is s-total in X. = The set of equivalence classes of M-N correspondences will be denoted by C(M, N), and we shall not make any distinction between a correspondence and its class. We shall write C(M) instead of C(M, M). There is a natural bijection A between C(M, N) and Corr(M, N), that will be described now. 1.4. Let X E C(M, N) and let HX = X ON L2(N) be the Hilbert space obtained by inducing the standard representation of N up to M via X ([22], Th. 5.1). The induced representation of M in HX and the right action of N on HX defined by give rise to an element A (X) = HX of Corr(M, N). Conversely, given H E Corr(M, N), let XH be the space HomNo(L2(N), H) of continuous N° -linear operators from L2 (N) into H. Let N acts on the right of X H by composition of operators and define on X H a N-valued inner product by r, s&#x3E; = r*s for r, s E XH. Then XH is a self-dual Hilbert N-module ([23], Th. 6.5). 336 Moreover, M acts on the left of XH by composition of operators and we obtain in this way a M-N correspondence. The maps X H HX and H H XH are inverse from each other ([4], Th. 2.2 and [23], Prop. 6.10). In fact, there is a natural isomorphism between the M-N correspondences X and HomN0(L2(N), X ~N L2(N)), given by assigning to any 03BE e X the element 039803BE:h ~ 03BE 0 h of HomNo(L2(N), X ~N L2(N)). 1.5. Let M, N, P be von Neumann algebras, X E C(M, N) and Y ~ C(N, P). We denote by X ON Y the self-dual completion (see [20], Th. 3.2) of the algebraic tensor product X 0 Y endowed with the obvious right action of P and the P-valued inner product LEMMA. (i) For x ~ .PN(X), there is an element p(x) in P(X ~N Y) well defined by We get in this way a normal homomorphism from 2N(X) into 2p(X ON Y). (ii) If the representation of N into 2p( Y) is faithful, then p is faithful. (iii) If we take Y = L2(N), viewed as an element of C(N, C), then p is an isomorphism of the von Neumann algebra 2N(X) onto the commutant HomNO(H x’ Hx) of the right action of N on Hx = X Q9NL2(N). Proof. For the proof of (i) see ([22], Th. 5.9 and [4], Prop. 2.9). Let us show that p is isometric under the assumption of (ii). If 03BE E X we define a continuous P-linear operator 0ç from Y into X ON Y by 0ç (~) = 03BE Q9 fi for fi E Y. It is easily checked that (039803BE)*(03BE’ ~ ~) = 03BE,03BE’)N~ for Ç’EX and ~ ~ Y, so that ~039803BE ~2 = ~0398*03BE 039803BE~ = ~03BE,03BE&#x3E;N~ = ~03BE~2. Let x ~ 2N(X) and e &#x3E; 0, and take j E X with Il 03BE Il = 1 and Il xç Il ~ x Il - s. Now choose fi E Y with Il ~ Il = 1 and 110xç(fI) ~ &#x3E; ~ xç Il - e. Then we have and from which it follows that ~ p(x) ~ = ~ x ~. Let us prove (iii) now. Obviously the range of p is contained in HomNo(Hx, Hx). Conversely, let r E HomNo(Hx, Hx) and consider the element r of 2N(X) such that 337 and thus p(Q = r. D 1.6. Keeping the notations of 1.5, we say that the self-dual Hilbert P-module X 0, Y provided with the homomorphism of M into 2p(X ON Y) given by restricting p is the composition correspondence of X by Y. It is the version in the setting of Hilbert modules of the composition of correspondences defined in ([8], §II). There are other classical operations on correspondences. We shall need the following ones. Let H E Corr(M, N) be a correspondence between M and N. Let H be the conjugate Hilbert space. If h E H, we denote by 11 the vector h when viewed as an element of H. Then H has a natural structure of correspondence from N to M by (see [21], 1.3.7). We call it the adjoint correspondence of H. Thanks to the bijection A between C(M, N) and Corr(M, N), we see that to each X E C(M, N) we can associate an element X E C(N, M), also called the adjoint correspondence of X. In general we haven’t an explicit description of X (see however 1.8 below). A subcorrespondence of X E C(M, N) is a submodule Y of X closed for the s-topology and stable by the left action of M. There is a natural bijection between the set of subcorrespondences of X and the set of projections in 2N(X) which commute with the range of M in 2N(X) by the left action. If X and Yare two M-N correspondences, we say that Y is contained in X and we write Y c X if Y is equivalent to a subcorrespondence of X. 1.7. We shall have to consider the following special case of composition of correspondences. Let H be a Hilbert space and N a von Neumann algebra. Then, in an obvious way, H is an element of C(2(H), C) and N is an element of C(C, N). Thus we may define the composition correspondence H Oc N, written H 0 N afterwards. The N-valued scalar product in H 0 N is given by Take an orthonormal basis (ei)ieI in H. Denote by 1;(1, N) the right N-module of 338 nets (yi)i~I of elements of N such that 03A3i~Iy*i yi is Q-weakly convergent. Provided with the N-module inner product (xi)i~I, (yi)i~I&#x3E; = LieIXr Yi’ it is a self-dual Hilbert N-module, and the map which sends (Yi)ieI on LieIei (8) yi is an isomorphism of Hilbert N-modules from l2w(I, N) onto H (8) N. (See [20], p. 457-459). We shall identify 1’(I, N) and 12(1) ~ N. Remark that 2N(H ~ N) may be identified to the von Neumann tensor product 2(H) (8) N in a natural way. 1.8. Next, we shall give fundamental examples of correspondences, related to completely positive maps. Let X E C(M, N) and 03BE E X. Then 03A6: x ~ 03BE, xç) is a completely positive normal map from M into N. We shall say that that (D is a coefficient of X, or is associated to X. Conversely, given a completely positive normal map (D from M into N, by the Stinespring construction we get a M-N correspondence XD. The self-dual Hilbert N-module is obtained by separation and self-dual completion of the right N-module MO N(algebraic tensor product) gifted with the N-module inner product The normal representation xo of M into N(X03A6) is given by If çcp denotes the class of 1 ~ 1 in X03A6, we have O(x) = 03BE03A6, x03BE03A6&#x3E; for each x E M, and 03BE03A6 is a cyclic vector for the correspondence XD. We shall say that Xq. is the correspondence associated to 03A6. If X is a M-N correspondence and 03BE is a cyclic vector in X, then it is easily seen that X is equivalent to the correspondence XD, where (D is the coefficient of X given by 03BE. Furthermore, every M-N correspondence is a direct sum of cyclic correspondences, so that, as pointed out by A. Connes in [8], the notions of completely positive maps and correspondences are closely related. When (D is a normal conditional expectation from M onto a von Neumann subalgebra N, it is easily checked that Xo is equivalent to the separated, self-dual completion of the right N-module M with N-valued inner product (m, m1) H 03A6(m*m1), endowed with the obvious left action of M. More generally, to every semi-finite normal operator valued weight C from M to N (see [ 14]), Def. 2.1 ), one can associate a M-N correspondence Xcp which extends the classical Gelfand- Segal construction for usual normal semi-finite weights (see [4], Prop. 2.8). The right M-module M endowed with its inner product m, m1&#x3E; m*m1 is = self-dual. Gifted with its natural left M-module structure, it is the M-M correspondence associated to the identity homomorphism of M. It will be called 339 the identity M-M correspondence, and denoted by XM or M; of course ^ (XM) = L2(M). Let now p be a normal homomorphism from M into a von Neumann algebra N. It is straightforward to show that X p is equivalent to the Hilbert N-subspace p( 1 )N of the right Hilbert N-module N, with left action of M given by Suppose next that N is a von Neumann subalgebra of M. The N-M correspondence associated to the inclusion z: N -+- M will be denoted by X N. Note that XN is obtained from XM = M by restricting to N the left action of M. Remark also that ^(XN) is L2(M) where we restrict to N the standard representation of M and keep the right action of M. Let E be a faithful normal conditional expectation from M onto N. It has been noticed in [4] that the (equivalence class of the) M-N correspondence X E is the adjoint correspondence XN of XN. Indeed, it is shown in ([4], Corol. 2.14) that A (XE) is equivalent to L2(M) considered as a M-N bimodule by restricting to N the right action of M, and this correspondence is easily seen to be equivalent to the adjoint of A (XN), thanks to the antilinear involutive isometry J of L2(M). (In fact, this remark remains true when E is any faithful normal semi-finite operator valued weight from M to N). Even if there doesn’t exist any conditional expectation from M onto N, we may consider XN. Note that by Lemma 1.5(iii), !£ N(X N) is isomorphic to the commutant of the right action of N on L2(M), since ^(XN) = L2(M) viewed as M-N bimodule. It follows that the normal homomorphism from M into N(XN) which appears in the definition of the M-N correspondence XN is injective, because it comes from the standard representation of M. The M-M correspondence X N ON X N will be denoted by YN. It has been introduced by Popa ([21], 1.2.4) in the finite case, as a very useful tool for the study of the inclusion N c M. When there exists a normal faithful conditional expectation E from M onto N, then YN = X E ON XN and YN is also the M-M correspondence associated to E viewed as a completely positive map from M to M (see [4], Th. 2.12). Let us remark that YM = X M = M. For N = C, the C-M correspondence Xc is the Hilbert M-module M with obvious action of C, and Xc is the Hilbert space L2(M) with the standard representation of M. Thus Yc = XC ~C xc L2(M) 0 M is the coarse M-M correspondence (see [8], Def. 3). 1.9. For later use, we prove the following result (see [21], Prop. 1.2.5.(ii)). LEMMA. Let M be a von Neumann algebra and N a finite dimensional von Neumann subalgebra of M. Then we have YN ce Yc. 340 Proof. Let z 1, ... , zk be the minimal projections of the centre Z(N), and (ejpq)1p,qnj a matrix units system for Nzj where j = 1,..., k. Let u p = ejp1 for p = ni and j = 1,..., k. We choose a normal faithful state 9 on M and we put ocj = (p(e i 1 1) for j = 1,..., k. Then one easily checks that the map E on M defined by is a normal faithful conditional expectation from M onto N. We take for L2(M) the standard form L2(M, ç) of the identity correspondence given by 9, and we identify M to a subspace of L2(M, ~). Let We have, for x ~ M, where 03A6 is E considered as a completely positive map from M to M. Thus, xçy 1-+’ xçCl»Y’ with x, y E M, induces an equivalence between the subcorrespon- dence of L2(M, ) ~ M having 03BE as cyclic vector and YN which is the M-M correspondence associated to 03A6. Notice that (D appears as a completely positive map which is a finite sum of completely positive maps factored by cp in the sense of ([19], Def. 1). 1.10. LEMMA. A correspondence X contains the identity correspondence M if and only if there exists a non zero central and separating vector 03BE in X (i.e. çx = xç for all x ~ M and if 03BEx = 0 then x = 0). Proof. The necessity of the existence of 03BE is obvious. Conversely suppose that there is a non zero separating central vector 03BE in X. Then 03BE, 03BE&#x3E; belongs to Z(M) and its support is 1. Consider the polar decomposition 03BE = ri«, ç)l/2 of 03BE (see [20], Prop. 3.11). Then 1 is central and since ~, ~&#x3E; is the support of (ç, ç), we have ~, ~&#x3E; = 1. Now it is easy to prove that ~M defines a subcorrespondence of X equivalent to M. D 341 1.11. REMARK. In ([21], Prop. 1.2.5) Popa has shown that for type III factors N c M the properties [M : N] oo and M c YN are closely related, where [M : N] denotes as usually the Jones’ index. More generally, let E be a faithful normal conditional expectation from a von Neumann algebra M onto a von Neumann subalgebra N. In [4], the index of E has been defined to be finite if there exists k &#x3E; 0 such that the map l 0 E - k|dM from M to M is completely positive (i being the injection of N into M). This definition is equivalent to the one given by Kosaki [18] when M and N are factors, and extends Jones’ definition. It follows easily from ([4] Th. 3.5) and Lemma 1.10 that M c YN when the index of E is finite, and that, conversely, if M c YN with N’ n M = C then the index of E is finite. Thus, Popa’s result remains true in general. 1.12. Recall that in [9] a topology has been defined on Corr (M, N), described by its neighbourhoods in the following way. DEFINITION. Let Ho E Corr(M, N), E &#x3E; 0, E c M and F c N two finite sets, and S = {h1,..., h J a finite subset of Ho. We denote by U(H0;03B5,E,F,S) the set of H E Corr(M, N) such that there exist kl, ... , kn E H with 1 ki, xkiy&#x3E; - hi, xhjy) 1 e for all x ~ E, y E F and i, j = 1,..., n. The we consider the well defined topology on C(M, N) for which these sets U are basis of neighbourhoods. Note that if we consider correspondences as representations of M Q9bin N° (the binormal ones), then it is easily verified that the above topology on Corr(M, N) is induced by the quotient topology introduced in [11] on the set of (unitary equivalence classes of) representations of M Q9bin N°. We shall now give an equivalent way of defining this topology on C(M, N). DEFINITION. Let X 0 E C(M, N), a 03C3-weak neighbourhood of 0 in N, E a finite subset of M and S = {03BE1, ..., 03BEn] a finite subset of X0. We denote by V(Xo; f, E, S) the set of X ~ C(M, N) such that there exist 11, .... rin E X with ~i, x~j&#x3E; - 03BEi, Xçj) E for all x ~ E and i, j = 1,..., n. We provide C(M, N) with the topology having such sets as basis of neighbourhoods. PROPOSITION. The bijection A : C(M, N) ~ Corr(M, N) is an homeomorphism. Proof. Let X0 E C(M, N) and Ho = X ° Q9N L2(N, cp), where 9 is a fixed faith- ful normal state on N. Denote by hep the canonical cyclic vector in L2(N, cp). Consider a neighbourhood U = U(H 0; e, E, F, S) of Ho. Then we may suppose that S = {03BE1 ~ h~,..., 03BEn ~ h~} with 03BE1,...,03BEn in X0, since the subspace {03BE Q9 hep, ç c- X. 1 is dense in Ho . Let: Then we shall prove that the image of V = V(Xo; , E, S’ ) by A is contained in U.