Some rigidity results for non-commutative Bernoulli shifts

We introduce the outer conjugacy invariants S( ), Ss ( ) for cocycle actions of discrete groups G on type II1 factors N, as the set of real numbers t >0 for which the amplification t of can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly S( ),Ss ( ) and the fundamental group of , F( ), in the case G has infinite normal subgroups with the relative property (T) (e.g., when G itself has the property (T) of Kazhdan) and is an action of G on the hyperfinite II1 factor by Connes–StZrmer Bernoulli shifts of weights {ti }i . Thus, Ss ( ) and F( ) coincide with the multiplicative subgroup S of R∗+ generated by the ratios {ti/tj }i,j , while S( ) = Z∗+ if S = {1} (i.e. when all weights are equal), and S( ) = R∗+ otherwise. In fact, we calculate all the “1-cohomology picture” of t , t >0, and classify the actions ( ,G) in terms of their weights {ti }i . In particular, we show that any 1-cocycle for ( ,G) vanishes, modulo scalars, and that two such actions are cocycle conjugate iff they are conjugate. Also, any cocycle action obtained by reducing a Bernoulliaction of a group G as above on N = ⊗g∈G(Mn×n(C), tr)g to the algebra pNp, for p a projection in N, p = 0, 1, cannot be perturbed to a genuine action. © 2005 Elsevier Inc. All rights reserved