Some rigidity results for non-commutative Bernoulli shifts

We introduce the outer conjugacy invariants L(sigma), L-s(sigma) for cocycle actions a of discrete groups G on type II1 factors N, as the set of real numbers t > 0 for which the amplification sigma(t) of sigma can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly L(sigma), L-s (sigma) and the fundamental group of sigma, F(sigma), 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 a is an action of G on the hyperfinite II1 factor by Connes-Stormer Bernoulli shifts of weights {t(i)}(i). Thus, L-s(sigma) and F(sigma) coincide with the multiplicative subgroup S of R+* generated by the ratios {t(i)/t(j)}(i,j), while L(sigma) = Z(+)* if S = {1} (i.e. when all weights are equal), and L(sigma) = R+* otherwise. 1n fact, we calculate all the "1-cohomology picture" of sigma(t), t > 0, and classify the actions (sigma, G) in terms of their weights {t(i)}(i). In particular, we show that any 1-cocycle for (sigma, G) vanishes, modulo scalars, and that two such actions are cocycle conjugate iff they are conjugate. Also, any cocycle action obtained by reducing a Bernoulli action of a group G as above on N = (x) over bar (g is an element of G) (M-n x n (C), tr)(g) to the algebra pNp, for p a projection in N, p not equal 0, 1, cannot be perturbed to a genuine action. (C) 2005 Elsevier Inc. All rights reserved.