Some rigidity results for non-commutative Bernoulli shifts

Abstract

We introduce the outer conjugacy invariants S ( σ ) , S s ( σ ) for cocycle actions σ of discrete groups G on type II 1 factors N, as the set of real numbers t > 0 for which the amplification σ t of σ can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly S ( σ ) , S s ( σ ) and the fundamental group of σ , F ( σ ) , 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 σ is an action of G on the hyperfinite II 1 factor by Connes–Størmer Bernoulli shifts of weights { t i } i . Thus, S s ( σ ) and F ( σ ) coincide with the multiplicative subgroup S of R + * generated by the ratios { t i / t j } i , j , while S ( σ ) = Z + * if S = { 1 } (i.e. when all weights are equal), and S ( σ ) = R + * otherwise. In fact, we calculate all the “1-cohomology picture” of σ t , t > 0 , and classify the actions ( σ , G ) in terms of their weights { t i } i . In particular, we show that any 1-cocycle for ( σ , 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 = ⊗ ¯ g ∈ G ( M n × n ( C ) , tr ) g to the algebra pNp, for p a projection in N, p ≠ 0 , 1 , cannot be perturbed to a genuine action.