Some rigidity results for non-commutative Bernoulli shifts

AbstractWe introduce the outer conjugacy invariants S(σ), Ss(σ) for cocycle actions σ of discrete groups G on type II1 factors N, as the set of real numbers t>0 for which the amplification σt of σ can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly S(σ),Ss(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 II1 factor by Connes–Størmer Bernoulli shifts of weights {ti}i. Thus, Ss(σ) and F(σ) coincide with the multiplicative subgroup S of R+* generated by the ratios {ti/tj}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 σt,t>0, and classify the actions (σ,G) in terms of their weights {ti}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(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