M. Gabriella Kuhn Università degli Studi di Milano

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1 AMENABLE ACTION AND WEAK CONTAINMENT OF CERTAIN REPREENTATION OF DICRETE GROUP M. Gabriella Kuhn Università degli tudi di Milano Abstract. We consider a countable discrete group Γ acting ergodically on a stard Borel space with quasi-invariant measure µ. Let π be a unitary representation of Γ on L 2 (, dµ, H) nicely related with (see definition below). We prove that if Γ acts amenably on then π is weakly contained in the regular representation. Let Γ be a countable discrete group acting (on the right) on a stard Borel space. uppose that the action is ergodic with µ as a quasi-invariant probability measure on. Let P(s, g) be the Radon-Nikodym cocycle of the action, that is P(s, g) = dµ(sg) dµ(s). Construct a representation of Γ as follows: (1) choose any Hilbert space H (2) choose any (Borel) cocycle A(s, g) : G U(H) where U(H) denotes the group of unitary operators on H. (3) construct H π = L 2 (, dµ, H) with the usaul inner product given by F, G = F(s), G(s) Hdµ(s). finally define (π(g)f)(s) = P(s, g) 1 2 A(s, g)f(sg). Observe now that although the group is not amenable it turns out that π is weakly contained in the regular representation for many natural constructions. In particular this is true when a) Γ is a free group is its Martin boundary (see [K Appendix]) b) Γ is a hyperbolic group is its boundary (see [A] use the techniques of [K ]) c) Γ is a lattice in a semisimple Lie group G = G/B is the maximal Furstenberg boundary of G (see [Q, sect.4]) Recall that a group is amenable if only if every unitary representation of Γ is weakly contained in π reg (here π reg denotes the regular representation) [H] [Z3 Proposition 7.3.6]. R. Zimmer introduced a class of Borel spaces which make a nonamenable group G look like an amenable one when acting on them. These spaces are called amenable G-spaces ( G is said to act amenably on them). To see that cases a) c) above deal with amenable actions of Γ on recall the following 1991 Mathematics ubject Classification. Primary 43A07, 22D10. Key words phrases. amenable actions, weak containment of representations. 1 Typeset by AM-TEX

2 2 M. GABRIELLA KUHN Proposition [Z3]. Let G be a locally compact group, H a closed subgroup Γ a lattice in G. Then Γ acts amenably on G/H if only if H is amenable. Case c) follows since minimal parabolic subgroups of semisimple Lie groups are amenable. For case a) the reader should think of the free group acting on a homogeneous tree as a discrete cocompact subgroup of the full automorphism group G of the tree. The Martin boundary of Γ is the set of all ends of the tree it was proved by C. Nebbia [N] that the stabilizer (in G) of an end is amenable. Case b) was proved by. Adams in [A]. In this paper we shall prove the following Theorem. uppose that a countable discrete group Γ acts ergodically on (, µ) with quasi-invariant measure µ. uppose that the action is amenable. Then every representation π constructed as above is weakly contained in π reg. We also believe that the converse is true, namely we believe that if for every Hilbert space H the above constructed representation (L 2 (, dµ, H), π) is weakly contained in π reg then the action is amenable. Unfortunatelly we were not able to prove this. We are happy to thank Tim teger for suggesting us the proof of Lemma 1. A special thank to Robert Zimmer for helpful conversations remarks for having given us [Z1] [Z2]. Proof of the result. Denote by π reg the right regular representation of Γ. Construct π reg π acting on l 2 (Γ) H π. Recall that U : l 2 (Γ) H π l 2 (Γ, H π ) given by (Uf)(x) = π(x)f(x) intertwines π reg π to π reg I, where I denotes the trivial representation. Thus π reg π is equivalent to a direct sum of copies of π reg it is sufficent to prove that π is weakly contained in π reg π. Fix any finite subset E of Γ. Let F L 2 (, dµ, H). Inside l 2 (Γ) L 2 (, dµ, H) we shall construct a sequence of functions (f n ) n=0 such that (piro(x))f n, f n approaches π(x)f, F for x in E. Identify l 2 (Γ) L 2 (, dµ, H) with L 2 (, dµ, l 2 (Γ) H). et f n (s) = φ n (s) F(s) where the φ n are elements of l 2 (Γ) L 2 (, dµ) = L 2 (, dµ, l 2 (Γ)). Compute π reg π(g)f n, f n = P(s, g) 1 2 πreg (g)φ n (sg), φ n (s) l 2 (Γ) A(s, g)f(sg), F(s) H dµ(s), π(g)f, F = P(s, g) 1 2 A(s, g)f(sg), F(s) H dµ(s). In order to prove our result we need a sequence of functions φ n (s, x) L 2 (, dµ, l 2 (Γ)) such that φ n(sg, xg)φ n (s, x) is weakly converging, in L 2 (, dµ), to the function identically one on. For amenable groups the existance of such a sequence is related to existance of an invariant mean in (L (G)). An analogue of the invariant mean property was proved for discrete groups by Zimmer in [Z1 Z2] in the case of an amenable action. The reader may take this as a definition of an amenable action:

3 Proposition 1. (Zimmer). uppose that Γ is a countable discrete group acting ergodically on (, µ). Then is an amenable Γ-space if only if there is a norm one linear map σ : L ( Γ) L () such that (1) σ(1) = 1, f 0 implies σ(f) 0; (2) If A is measurable, σ(f χ p 1 A) = σ(f)χ A where p : Γ is the projection; (3) σ(f g) = σ(f) g, where f g(s, x) = f(sg, xg) φ g(s) = φ(sg). In [Z1] it is shown how to construct σ once we know that the action of Γ on is amenable. From these we may deduce other properties of σ which will be needed. For the reader s convenience we briefly recall the construction. Let Ũg : L 2 ( Γ) L 2 ( Γ) be defined by 3 (Ũgf)(s, x) = f(sg, xg)p(s, g) 1 2. For f L () define a multiplication operator M f on L 2 ( Γ) by letting (M f h)(s, x) = f(s)h(s, x). Let R denote the von Neumann algebra generated by {Ũg, M f } in B(L 2 ( Γ)). It was proved in [Z2] that, if Γ acts amenably on then there is a norm one projection hence a conditional expectation P : B(L 2 ( Γ)) R. Consider now the following decomposition for L 2 ( Γ). For f L 2 ( Γ) let f s (x) = f(sx, x)p(s, x) 1 2. Then f f sdµ is a unitary isomorphism of L 2 ( Γ) l2 (Γ)dµ. Further every element of R is decomposable with respect to this direct integral decomposition. (Namely Ũg corresponds to π reg(g) M f to fs where (f s h s )(x) = f(sx)h s (x)). For every f L ( Γ) we have the multiplication operator M f B(L 2 ( Γ)). Now P(M f ) is decomposable, hence we can write P(M f ) = T s f dµ. Our map σ is now given by : (σf)(s) = T f s (δ e), δ e l2 (Γ), where δ e denotes the characteristic function of the identity e. In particular, if F is an element of L () we let F(s, g) = F(s) for every g Γ then i) (σf)(s) = F(s) furthermore, for every h L ( Γ) ii) (σhf)(s) = (σh)(s)f(s), since P(M h M F ) = P(M h )M F because F is an element of R. In order to use our function σ we need the Radon-Nikodym derivative P(s, g) 1 2 to be a bounded function of s for any fixed g. This, in general may or may not be true. However we shall prove that it is always possible to find a measure ν equivalent to µ, for which this is true.

4 4 M. GABRIELLA KUHN Lemma 1. Let Γ be a discrete group. There exists a length function x x such that the cardinality of the sets { x N} is less than or equal to 3 N. Given such a length function choose K > 3 set ν(e) = K x µ x (E) where µ x (E) = µ(ex) for every measurable E. Then ν is equivalent to µ the Radon-Nikodym derivative dν(sg) dν(s) is less than or equal to K g. Proof. Choose any ordering {x n } n=0 for the elements of Γ satisfying x 0 = e. et x = inf{ J n j : x = x ± n 1 x ± n 2...x ± n J }. j=1 It is immediate that is a length function. et Φ(N) = {x : x = N}. We observe that the number of strictly positive solutions (n 1, n 2,..., n k ) of the equation n 1 + n n k = N is ( N 1 k 1). Hence Finally Φ(N) N k=1 ( ) N 1 2 k = 2 3 N 1. k 1 {x : x N} N 1 = 3 N. It is obvious from the definition of ν from the quasi-invariance of µ that ν µ are equivalent measures. Let E be a measurable set. Compute ν(eg) = K x µ(egx) = K x g + g µ(egx) K g K gx µ(egx) = K g ν(e). This completes the proof. From this point on we shall assume that the Radon-Nikodym derivative P(s, g) is a bounded function of s for any given g. Proof of the Theorem. Let P = {φ L 1 ( Γ) φ(s, x) 0 φ(s, x)dµ(s) = 1}. First af all observe that P is weak- dense in the unit ball of (L ( Γ)). Construct a linear functional σ on L ( Γ) by letting σ (f) = (σf)(s)dµ(s). Recall that (h g)(s, x) = h(sg, xg) (ψ )g(s) = ψ(sg).

5 5 Choose: h in L ( Γ), ǫ positive g in a finite set E of Γ. First of all we shall show that it is possible to find φ L 1 ( Γ) such that ( ) (1.1) (φ(sg, xg) φ(s, x))h(s, x) dµ(s) < ǫ for every g in E. In fact, by property c) of Proposition 1, we have (σh)(s)dµ(s) = (σh g 1 )(s)p(s, g 1 )dµ(s) because of ii) = (σh g 1 P(s, g 1 ))(s)dµ(s) Hence it is possible to choose φ L 1 ( Γ) so that ( ) (1.2) (σh)(s) φ(s, x)h(s, x) dµ(s) < ǫ (1.3) ( ) (σh g 1 P(s, g 1 ))(s) φ(s, x)h(sg 1, xg 1 )P(s, g 1 ) dµ(s) < ǫ But ( ) φ(s, x)h(sg 1, xg 1 )P(s, g 1 ) dµ(s) = P(s, g 1 ) φ(s, xg)h(sg 1, x)dµ(s) letting sg 1 = s = φ(sg, xg)h(s, x)dµ(s). ubtracting (1.3) from (1.2) we get (1.1) Moreover, since σ(1) = 1, we may also require that, for any given h L () (1.4) h (s)(1 φ(s, x))dµ(s) < ǫ. We shall use now Namioka s argument to pass from the weak to the strong closure. Let F = L 1 () g Γ L1 ( Γ) = F 0 g Γ F g where L 1 () each copy of L 1 ( Γ) has the norm topology. The weak topology on F is the product topology for the weak topology on L 1 () on L 1 ( Γ). Define a linear map T : P F by letting ( ) T(φ) = φ(s, x) 0 ( ) T(φ) = φ(sg, xg) φ(s, x). g

6 6 M. GABRIELLA KUHN Then (1.1) (1.4) say that the point (1, 0,...,0...) is in the weak closure of T(P) since T(P) is convex, it follows that the same point is in the strong closure of T(P). Hence we can find a net of elements ψ j P such that ψ j (s, x) 1 ψ j (sg, xg) ψ j (s, x) 0 strongly respectively in L 1 () in L 1 ( Γ). Finally let φ j (s, x) = ψ j (s, x). By stard arguments φ j g φ j L2 ( Γ) ψ j ψ j 1 2 L 1 ( Γ). Choose any F in L 2 (, dµ, H). et f n (s, x) = φ n (s, x) F(s). Let G(s, g) = P(s, g) 1 2 A(s, g)f(sg), F(s) H. ince the set of functions F : H such that (π(g)f)(s), F(s) is a bounded function of s is dense in L 2 (, dµ, H), we may assume that G(s, g) is a bounded function of s for any given g Γ. We may also assume that G(s, g) 0. Observe that G(s, g)(φ n g φ n ) L 2 ( Γ) 0. We have G(s, g)(φ n g φ n ) L2 ( Γ) = G(s, g) ψ n (sg, xg)dµ(s)+ G(s, g) ψ n (s, x)dµ(s) 2 G(s, g) (φ n (sg, xg)φ n (s, x))dµ(s). But ψ n(s, x) converges to 1 in L 1 () so does ψ n(sg, xg) since ψ n g ψ n L1 ( Γ) 0. Hence G(s, g) (φ n (sg, xg)φ n (s, x))dµ(s) G(s, g)dµ(s). References A.. Adams, Boundary amenability for hyperbolic groups an application to smooth dynamic of simple groups, preprint. H. A. Hulanicki, Means Folner conditions on locally compact groups, tudia Math. 27 (1966), K. M.G. Kuhn, T. teger, More irreducible boundary representations of free groups, preprint. N. C. Nebbia, Amenabilty Kunze tein property for groups acting on a tree, Pacific J. Math. 135 (1988), Q. J.C. Quigg, J. pielberg, Regularity hyporegularity in C -dynamical systems, Houston J. Math. 18 (1992), Z1. R.J. Zimmer, On the von Neumann algebra of an ergodic group action, Proc. Amer. Math. oc. 66 (1977), Z2., Hyperfinite factors amenable ergodic actions, Invent. Math. 41 (1977), Z3., Ergodic Theory emisimple Groups, Birkhäuser, Boston, Dipartimento di Matematica Università, via C. adini Milano