We 64257rst show that coamenability does not pass to subgroups answering a question asked by Eymard in 1972 We then address coamenability for von Neumann algebras describing notably how it relates to the former esum e Nous d57524emontrons tout dabor ID: 35403 Download Pdf

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We 64257rst show that coamenability does not pass to subgroups answering a question asked by Eymard in 1972 We then address coamenability for von Neumann algebras describing notably how it relates to the former esum e Nous d57524emontrons tout dabor

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ON CO-AMENABILITY FOR GROUPS AND VON NEUMANN ALGEBRAS NICOLAS MONOD & SORIN POPA Abstract. We ﬁrst show that co-amenability does not pass to subgroups, answering a question asked by Eymard in 1972. We then address co-amenability for von Neumann algebras, describing notably how it relates to the former. esum e. Nous demontrons tout d’abord que la comoyennabilite ne passe pas aux sous-groupes, resolvant ainsi une question posee par P. Eymard en 1972. Nous etudions ensuite la comoyennabilite pour les alg`ebres de von Neumann

; en particulier, nous clariﬁons le lien avec le cas des groupes. Co-amenablesubgroups A subgroup of a group is called co-amenable in if it has the following relative ﬁxed point property: Every continuous aﬃne -action on a convex compact subset of a locally convex space with an -ﬁxed point has a -ﬁxed point. This is equivalent to the existence of a -invariant mean on the space G/H ) or to the weak containment of the trivial representation in G/H ); see [4] (compare also Proposition 3 below). Alternative terminology is amenable pair or co-Flner , and

more generally Greenleaf [5] introduced the notion of an amenable action (conﬂicting with more recent terminology) further generalized by Zimmer [10] to amenable pairs of actions. In the particular case of a normal subgroup , co-amenability is equivalent to amenability of the quotient G/H One checks that for a triple K co-amenability of in and of in implies that is co-amenable in . In 1972, Eymard proposed the following problem [4, N 8]: If we assume co-amenable in , is it also co-amenable in (It is plain that is co-amenable in .) If is normal in , then the answer is positive, since it

amounts to the fact that amenability passes to subgroups. We give an elementary family of counter-examples for the general case. Interestingly, we can even get some normality: Theorem 1. (i) There are triples of groups with co-amenable in but not in (ii) For any group there is a triple with co-amenable in and /K Moreover, all groups in (i) can be assumed ICC (inﬁnite conjugacy classes). In (ii), the group is ﬁnitely generated/presented, ICC, torsion-free, etc. as soon as has the corresponding property. Remark . We shall see that one can also ﬁnd counter-examples K < H < G

with all three groups ﬁnitely generated. 2000 Mathematics Subject Classiﬁcation. 43A07, 43A85, 46L.

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2NICOLASMONOD&SORINPOPA Remark . We mention that in a recent independent work V. Pestov also solved Eymard’s problem (see V. Pestov’s note in this same issue). We start with the following general setting: Let be any group and a monomorphism. Denote by the corresponding HNN-extension, i.e. def K,t tkt Write def Kt so that Proposition 2. The group is co-amenable in Proof. Note that is co-amenable in , so that by applying the relative ﬁxed point property to the

space of means on G/K ) we deduce that it is enough to ﬁnd an -invariant mean G/K For every we deﬁne a mean by ) = ) for in G/K ). This mean is left invariant by the group Kt . Since is the increasing union of Kt for , any weak-* limit of the sequence is an -invariant mean. We set Kt and consider K < H < H . Observe that is not co-amenable in unless ) is co-amenable in , and hence also not in when . Thus we will construct our counter-examples by ﬁnding appropriate pairs ( K, ) with ) not co-amenable in For Theorem 1 , let be any group, let be the direct limit of

ﬁnite Cartesian powers and let be the shift; then ) is co-amenable in if and only if is amenable. The claims follow; observe that is the restricted wreath product of by For Remark 1 , we may for instance take to be the free group on two generators and deﬁne by choosing two independent elements in the kernel of a surjection of to a non-amenable group. Amenability of groups can be characterized in terms of bounded cohomology ; see [6]. Here is a generalization, which shows that actually the transitivity and (lack of) hereditary properties of co- amenability were to be expected.

Proposition 3. The following are equivalent: (i) is co-amenable in (ii) For any dual Banach -module and any , the restriction G,V H,V is injective. Proof. “( ii )” is obtained by constructing a transfer map (at the cochain level) using an invariant mean on G/H , along the lines of [7, 8.6]. “( ii )” Let be the -module G/H , where is embedded as constants. The extension 0 G/H 0 splits over (by evaluation on the trivial coset) so that in the associated long exact sequence the transgression map H,V ) vanishes. By (ii) and naturality, the transgression G,V ) also vanishes, and so the long exact

sequence for yields -invariant element in G/H which does not vanish on the constants. Taking the absolute value of (for the canonical Riesz space structure) and normalizing it gives an invariant mean. Co-amenablevonNeumannsubalgebras Let be a ﬁnite von Neumann algebra and a von Neumann subalgebra. For simplicity, we shall assume has countably decomposable centre. Equivalently, we assume has a normal faithful tracial state . Let N,B be the basic construction for N, i.e. N,B JBJ ∩B N, with the canonical conjugation on N, ) and the canonical projection of N, ) onto B, ). Note that,

up to isomorphism, the inclusion N,B does not depend on

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CO-AMENABILITYFORGROUPSANDVONNEUMANNALGEBRAS3 . We also denote by Tr the unique normal semiﬁnite faithful trace on N,B given by Tr( xe ) = xy ,x,y Deﬁnition 4 ([8], see also [9]). The subalgebra is co-amenable in if there exists a norm one projection of N,B onto . One also says that is amenable relative to Remark . If and is amenable relative to then is amenable relative to Indeed, since N,B is included in N,B , we may restrict Ψ : N,B to N,B The next result from [8, 9] provides several alternative

characterizations of relative amenability. The proof, which we have included for the reader’s convenience, is an adaptation to the case of inclusions of algebras of Connes’ well known results for single von Neumann algebras [3]. Proposition 5. With N, as before, the following conditions are equivalent: (i) is co-amenable in (ii) There exists a state on N,B with uXu ) = for all N,B ∈ U (iii) For all ε > and all ﬁnite ⊂ U there is a projection in N,B with Tr f < such that ufu Tr < Tr for all Here (ii) parallels the invariant mean criterion while (iii) is a Flner

type condition. Proof. “( ii )” If Ψ is a norm one projection onto , then by Tomiyama’s Theorem it is a conditional expectation onto ; thus Ψ is Ad( )-invariant ∈ U ). Conversely, if satisﬁes ( ii ) above and N,B then let Ψ( ) denote the unique element in satisfying (Ψ( . It is immediate to see that 7 Ψ( ) is a conditional expectation (thus a norm one projection) onto “( ii iii )” Assuming ( iii ), denote by the set of ﬁnite subsets of ) ordered by inclusion and for each let ∈ P N,B ) be a non-zero projection with Tr( such that uf Tr Tr . For

each N,B let def = lim Tr( Xf Tr( ) for some Banach limit over . Then is easily seen to be an Ad( )-invariant state on N,B ∈ U ). Conversely, if satisﬁes ( ii ) then by Day’s convexity trick it follows that for any ﬁnite ⊂ U and any ε> 0 there exists N,B Tr) such that Tr( ) = 1 and uηu Tr . By the Powers-Strmer inequality, this implies u Tr Tr . Thus, Connes’ “joint distribution” trick (see [2]) applies to get a spectral projection of such that ufu Tr Tr The next result relates the co-amenability of subalgebras with the co-amenability of subgroups,

generalizing a similar argument for single algebras/groups from [3]. Proposition 6. Let be a (discrete) group, , a ﬁnite von Neumann algebra with a normal faithful tracial state and Aut( , a trace-preserving cocycle action of on , . Let be the corresponding crossed product von Neumann algebra with its normal faithful tracial state given by ) = . Let also H be a subgroup and Then is co-amenable in if and only if the group is co-amenable in In particular, taking , we deduce: Corollary 7. Let H < G be (discrete) groups. Then the inclusion of group von Neumann algebras is co-amenable if and

only if is co-amenable in Proof of Proposition 6. ” Let G/H be a net of ﬁnite Flner sets, which we identify with some sets of representatives . For each and N,B set ) =

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4NICOLASMONOD&SORINPOPA where Φ : sp Ne is the canonical operator valued weight given by Φ( xe ) = xy for x,y Then clearly are completely positive, normal, unital, -bimodular and satisfy ) = , where gF . Thus, if we put Ψ( def = lim ) for some Banach limit over , then Ψ( ) = for all , Ψ( N,B and Ψ is completely positive and unital. Thus Ψ is a norm one

projection. ” Let be a set of representatives for the classes G/H and denote by A N,B the set of operators of the form , where , sup . Clearly, G/H ). Note that Ad( ) implements the left translation by on G/H ) via this isomorphism. Put def ; then is clearly a left invariant mean on G/H ). Corollary 8. There are inclusions of type II factors such that is co-amenable in but not in . In fact, there are such examples with having property (T) relative to in the sense of [1, 8, 9] Proof. Let be an inﬁnite group with property (T) and let < G be as in point (ii) of Theorem 1. Set ), ), ).

Choosing to be ICC ensures that these three algebras are factors of type II . The claim follows now from Corollary 7. We proceed now to present an alternative example of interesting co-amenable inclusions of type II factors. Let be a free ultraﬁlter on . Denote as usual by the hyperﬁnite II factor and let def N,R /I , where def lim ) = 0 . Then is well known to be a type II factor with its unique trace given by (( ) = lim ). Let further , where is embedded into as constant sequences. Recall that are factors and ( Also, if is an automorphism of then there exists a unitary element

normalizing both and such that Ad( . The unitary element is unique modulo perturbation by a unitary from and is properly outer if and only if Ad( ) is properly outer on (see Connes [2] for all this). Thus, if is a properly outer cocycle action of some group on then Ad( } ⊂ N ) implements a cocycle action of on (again, see [2]). Theorem 9. (i) is amenable both relative to and to (ii) If is the von Neumann algebra generated by the normalizer of in , then is not co-amenable in . Moreover, if is a properly outer action of an inﬁnite property (T) group on e.g. by Bernoulli shifts) then

def = ( ∪ { 00 is not amenable relative to . In fact, has property (T) relative to Proof. By Remark 3 it is suﬃcient to prove that is amenable relative to . We show that in fact a much stronger version of condition (iii) in Proposition 5 holds true, namely: ∀U ⊂ U countable, ∈ U ) such that ve satisﬁes ufu f, ∈ U or, equivalently, Let = ( with ∈ U k,n . Let ) and for each let be such that /n, k,m . Thus, if we put def = ) then ,R . In particular, . For each let now be a unitary element satisfying )) (this is trivially possible, since the latter

is a type II factor). Thus, if we let = ( then vPv , showing that as well. The rest of the statement is now trivial, because if we denote by the quotient group ) then itself is a cross product of the form ( . By Proposition 6 (with and = 1), the co-amenability of in amounts to the amenability of the group But contains copies of any countable non-amenable group ( e.g. property (T) groups).

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CO-AMENABILITYFORGROUPSANDVONNEUMANNALGEBRAS5 References [1] C. Anantharaman-Delaroche. On Connes’ property for von Neumann algebras. Math. Japon. , 32(3):337–355, 1987. [2] A. Connes.

Classiﬁcation of injective factors. Cases II ,II ,III , = 1. Ann. of Math. (2) , 104(1):73–115, 1976. [3] A. Connes. On the classiﬁcation of von Neumann algebras and their automorphisms. In Symposia Mathematica, Vol. XX (Convegno sulle Algebre e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria , INDAM, Rome, 1975) , pages 435–478. Academic Press, London, 1976. [4] P. Eymard. Moyennes invariantes et representations unitaires . Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 300. [5] F. P. Greenleaf. Amenable

actions of locally compact groups. J. Functional Analysis , 4:295–315, 1969. [6] B. E. Johnson. Cohomology in Banach algebras. Mem. Am. Math. Soc. , 127, 1972. [7] N. Monod. Continuous bounded cohomology of locally compact groups . Lecture Notes in Mathematics 1758, Springer, Berlin, 2001. [8] S. Popa. Correspondences. INCREST preprint, 1986. [9] S. Popa. Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property T. Doc. Math. , 4:665–744 (electronic), 1999. [10] R. J. Zimmer. Amenable pairs of groups and ergodic actions and the associated

von Neumann algebras. Trans. Amer. Math. Soc. , 243:271–286, 1978. N.M.: University of Chicago, Chicago IL 60637 USA E-mail address monod@math.uchicago.edu S.P.: UCLA, Los Angeles CA 90095 USA E-mail address popa@math.ucla.edu

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