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Pacific Journal of Mathematics

COHOMOLOGY OF ACTIONS OF DISCRETE GROUPS ON FACTORS OF TYPE II1 YASUYUKI K AWAHIGASHI

Vol. 149, No. 2

June 1991

PACIFIC JOURNAL OF MATHEMATICS Vol. 149, No. 2, 1991

COHOMOLOGY OF ACTIONS OF DISCRETE GROUPS ON FACTORS OF TYPE Hi YASUYUKI KAWAHIGASHI We study 1-cohomology of discrete group actions on factors of type Πi Characterizations of Kazhdan's property T and amenability for discrete groups in terms of cocycles and coboundaries are given, and we show that each of SL(Λ , Z ) , n > 3 , and Sp(«, Z ) , n > 2, has a continuous family of mutually non-cocycle conjugate free actions on the AFD factor of type II i as an application. We also introduce and compute entropy for discrete amenable group action on factors of type Hi.

0. Introduction. In this paper, we study 1-cohomology of discrete group actions on factors of type II i . We give characterizations of Kazhdan's property T and amenability for discrete groups in terms of 1-cocycles and coboundaries for actions on factors of type II i . As an application, we also show that each of SL(n, Z ) , n > 3, and Sp(π, Z ) , n > 2, has a continuous family of mutually non-cocycle conjugate ergodic free actions on the approximately finite dimensional (AFD) factor of type II i . These are typical groups with Kazhdan's property T. We introduce and compute entropy of discrete amenable group actions on the AFD factor of type II i . Complete classification of actions of discrete amenable groups on the AFD factor 31 of type IIi up to cocycle conjugacy was given in Ocneanu [O]. In particular, he showed that any two free actions of a discrete amenable group on 31 are cocycle conjugate. Then Jones [J2] showed that this statement is no longer valid for any discrete nonamenable group. He constructed two free actions and used the ergodicity at infinity to distinguish the two. This shows that non-amenable discrete groups are quite different from amenable ones in the theory of group actions on factors. In order to understand cocycle conjugacy of nonamenable group actions, we start to study 1-cohomology of the actions and get several von Neumann algebra analogues of Schmidt's work [S] on ergodic actions on probability spaces. A major difference between the cohomology theory on probability spaces and one on von Neumann algebras is that we do not have the group structure on the space of cocycles in the latter case, which causes technical difficulty.

303

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YASUYUKIKAWAHIGASHI

First we work on groups with Kazhdan's property T, which are far from being amenable. (See §1 for the definition.) Since Connes [C3], several authors have shown that the discrete groups with Kazhdan's property T are the opposite extreme of amenable groups with respect to representations and actions on probability spaces and von Neumann algebras. Here a characterization of Kazhdan's property T in terms of cocycles and coboundaries is given in §1. Jones [Jl] showed two mutually non-cocycle conjugate free actions of discrete groups with Kazhdan's property T on the AFD factors of type I I i , and M. Choda [Ch2] showed four of such actions. We exhibit a continuous family of mutually non-cocycle conjugate free actions on the AFD factor of type Hi for certain groups with Kazhdan's property T like SL(n, Z ) , n > 3 and Sp(n,Z), ft > 2, using her construction in [Ch2]. These cannot be distinguished by ergodicity at infinity as in Jones [J2]. Instead, we make use of the rigidity argument of cocycles to show that the "almost all" pair in the family is not mutually cocycle conjugate. This shows another aspect of rigidity in operator algebras. (See Connes [C3], Connes-Jones [CJ1, CJ2].) In §2, we work on amenable groups. We introduce and compute Connes-Stormer entropy for discrete amenable group actions on the AFD factor of type II i to distinguish continuously many non-commutative Bernoulli shifts. Then we get a characterization of amenability in terms of cocycles and coboundaries, based on Ocneanu's work [O]. This shows another remarkable difference between the discrete groups with Kazhdan's property T and discrete amenable groups. The author is thankful to Professor M. Choda for helpful conversations and to Professor M. Takesaki for a suggestion improving the exposition. 1. Kazhdan's property T and cohomology classes. Let G be a discrete (countable) group and a an action of G on a factor Jt of type I I X . Let Zl = {u:G~+ %ί(^)\ugag(uh) ι

= ugh,

g,heG},

fί{Jt)\ there exists υ e &(JT)

B a = {u:G-+

such that ug =

vag(v*)}.

Topology of Z\ is given by the strong convergence at each g e G. This topology is given by the following metric: 00

d(u, v) = Σ

j

2n\\ugn - vgn\\2,

Consider the induced metric on B^.

where G = {gn\n > 1}.

COHOMOLOGY OF ACTIONS OF DISCRETE GROUPS

305

We say two cocycles ug, υg are cohomologous and write ug ~ vg if there exists a unitary u e Jt such that ug = wι;gα^(w*). Let H£ = Z\l ~ be the set of cohomology classes. Araki-Choda [ACh] called an action a strong if any asymptotically α-fixed sequence in a Jt is equivalent to a bounded sequence in Jt and showed that a discrete group has Kazhdan's property T if and only if every action of it on a factor of type II i is strong. This is a non-commutative analogue of Connes-Weiss [CW] to the effect that a discrete group has Kazhdan's property T if and only if every ergodic measure-preserving action of it on a probability space is strongly ergodic. We show that strong actions have a certain good property about cocycles. The following technique is based on that of Connes [Cl, Theorem 3.1], and we note an analogy between central sequences and asymptotically α-fixed sequences. PROPOSITION 1. If a is strong in the above context, then the space of coboundaries B^ is closed.

Proof We define a bijective map Φ from the coset space %(J?)l%i{jra) to Z\ by Φ(υ) = {vag(v*)}. Because the metric defined by ZΛnorm is translation invariant, %(J?)l%f{Jra) is a Polish space. We show that this Φ is a homeomorphism. If Vn^ίJT") -• v^{jra) in the topology of the coset space, then there exists a sequence of unitaries (wn) in %(Jίa) such that \\vnwn - v\\ -* 0. This implies \\vnag(v*) - vag(υ*)\\2 —• 0, and hence Φ is continuous. On the other hand, suppose vnag(υ*) —• υa(υ*). Then the sequence (v*vn) is asymptotically α-fixed; hence there exists a sequence (wn) in Jί* such that \\v*vn-wn\\2 —• 0. We may assume each wn is a unitary by, changing wn by a small operator if necessary, by Proposition 1.1.3 in Connes [C2]. Then the cosets vn^{Jίa) converge to v%{jra), 1 hence Φ " is also continuous. Thus there exists a positive number en such that d(uag(u*), 1) < εn implies existence of w £ ^{J^a) with \\uw — 1||2 < 1/2W. Suppose we have a sequence (vnag(v*)) converging to ug in Z\ . Choosing a subsequence if necessary, we may assume d(vnag(υ*), vn+ιag(v*+1)) < en. We choose a sequence of α unitaries (un) in ^ ( ^ # ) by induction so that \\vnun-υn+\un+ι\\2< \/2n . Suppose u\, ... , un are chosen. Then there exists a unitary a W w e %ί{J? ) such that \\v*+ιvnwn+ι - 1|| 2 < 1/2 by the definition of εn. Now set un+\ = unwn+\. The sequence {vnun) converges to some unitary M G / . We then have ug = υag(v*). D

306

YASUYUKIKAWAHIGASHI

Here we recall the definition of Kazhdan's property T for later use. A discrete group G is said to have Kazhdan's property T if it has the following condition: There exists a finite subset F cG and a positive number ε such that for any unitary representation Ug of G on H, if there exists a vector ξ e ^ such that \\ξ\\ = 1 and \\Ugξ -ξ\\ 0 /w/ ,/or any pair of von Neumann subalgebras N, P c &, the

[THEOREM

conditions dim N = n and N c P imply H(N\P) < ε. Let G be a countable amenable group. Choose an increasing sequence of finite subsets {An} by amenability such that Λ-00

|Λ|

(It is well known that existence of such a sequence is equivalent to amenability. See Greenleaf [G] for instance.) For an action a of G on a factor Jt of type II i , we define =

H{agι(N),...,agm(N))9 where A = {gu ... ,

gm}cG,

COHOMOLOGY OF ACTIONS OF DISCRETE GROUPS

313

a) = lim sup γ-r-rH(N, a, An), «->oo \Λn I #(α) = sup H(N,a).

H(N

9

NCJT, finite dimensional

Note that # ( # , α) < H(N) by property (B) in Connes-Stormer [CS]. The sup in the third definition may be infinity. The following is the non-commutative Kolmogorov-Sinai theorem and an analogue of Theorem 2 in [CS]. 12. Let Pq, q > 1, be an increasing sequence of finite dimensional subalgebras of the AFD factor 31 of type Hi \^Fq=3ί. Then PROPOSITION

H(a)= Proof. The same proof as that of Theorem 2 in [CS] works. (Use property (F) and Theorem 1 in [CS] to get H(N, a) < H(Pq 9a) + e for given e.) D The following is a computation of entropy for Bernoulli shifts and corresponds to Theorem 3 in [CS]. 13. Represent the AFD factor 3ί of type Hi as the infinite tensor product of k-dimensional matrix algebra Mk(C) with respect to the trace over a countable amenable group G. Define an action ag of G on 31 by the Bernoulli shift Then we get H(a) = log/:. PROPOSITION

Proof. For q > 1, define g€Aq

q

It is clear that Pq is an increasing sequence of finite dimensional algebras in 31 and |J£Li Pq is weakly dense in 31. Thus we can apply Proposition 12, and get H{μ) = lim lim

τ^H{Pq,a,An) \An\

= log/c, by the definition of An .

314

YASUYUKI KAWAHIGASHI

The following shows that this entropy for group actions is more powerful than entropy of single automorphisms. 14. Let G = Z 2 and apply the above construction for k = 2, 3 to get actions a and β of Z 2 on 3ί. Then for any g e 7?, g Φ 0, ag and βg have the entropy infinity as single automorphisms. Actually, they are both conjugate to the shift on ® z ^ = 31. But as group actions, they have different entropy, and thus they are nonconjugate. Let G = Z[l/2]/Z and apply the above construction for k = 2, 3 to get actions a and β of Z[l/2]/Z on 3ί. Then for any g e Z[l/2]/Z, α g and βg have the entropy zero as single automorphism because they are both periodic, by Remark 6 of Connes-Stormer [CS]. But as group actions, they have different entropy, and thus they are non-conjugate. EXAMPLE

We would like to get continuously many values of the entropy, so we introduce the following as in Theorem 4 in [CS]. Let Jί be the infinite tensor product of Af^(C) with respect to the product state ψλ = 0 φλ, λ = (λ\, ... , λ^), where φλ on M^(C) is defined by •..

,

λ1 + - " + λ * = l,

λj>0.

Let βg be the ergodic action of a countable amenable group G on Jί given by the Bernoulli shift. The centralizer Jt^ is isomorphic to 31 as in the proof of Theorem 4 of Connes-Stormer [CS]. Define the action aλg to be the restriction of βg on / i . This is also ergodic, and this is a free action. We use entropy as a conjugacy invariant to distinguish these actions. PROPOSITION

15. The entropy H(aλ) of the above action a\ is given

by k 7=1

Proof. Define

COHOMOLOGY OF ACTIONS OF DISCRETE GROUPS

315

Then using Properties (C), (D), (E) and (C) of Connes-Stermer [CS], we get

H(Fp,a,An), Kazhdan 's property T, strong ergodicity and invariant means for ergodic group-actions, Ergodic Theory Dynamical Systems, 1 (1981), 223-226. R. J. Zimmer, On the cohomology of ergodic actions ofsemisimple Lie groups and discrete subgroups, Amer. J. Math., 103 (1981), 937-950. , Ergodic Theory and Semisimple Groups, Birkhauser, 1984.

Received January 4, 1990. UNIVERSITY OF TOKYO HONGO, TOKYO, 113 JAPAN

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Pacific Journal of Mathematics Vol. 149, No. 2

June, 1991

Manuel Alfaro Garcia, Mark Conger and Kenneth Hodges, The structure of singularities in 8-minimizing networks in R2 . . . . . . . . . . . . . . . . . . . . . . 201 Werner Balser, Dependence of differential equations upon parameters in their Stokes’ multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Enrico Casadio Tarabusi and Stefano Trapani, Envelopes of holomorphy of Hartogs and circular domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Hermann Flaschka and Luc Haine, Torus orbits in G/P . . . . . . . . . . . . . . . . . 251 Gyo Taek Jin, The Cochran sequences of semi-boundary links . . . . . . . . . . . . . 293 Yasuyuki Kawahigashi, Cohomology of actions of discrete groups on factors of type II1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Ki Hyoung Ko and Lawrence Smolinsky, A combinatorial matrix in 3-manifold theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 W. B. Raymond Lickorish, Invariants for 3-manifolds from the combinatorics of the Jones polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Peter Arnold Linnell, Zero divisors and group von Neumann algebras . . . . . . 349 Bruce Harvey Wagner, Classification of essential commutants of abelian von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Herbert Walum, Multiplication formulae for periodic functions . . . . . . . . . . . . 383