Some rigidity results for non-commutative Bernoulli shifts

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Abstract

We 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=¯gG(Mn×n(C),tr)g to the algebra pNp, for p a projection in N, p0,1, cannot be perturbed to a genuine action.

Keywords

Cocycles
Bernoulli actions
Property (T) groups

Cited by (0)

A preliminary version of this paper was circulated as MSRI preprint No. 2001–2005 under the title “A rigidity result for actions of property (T) groups by Bernoulli shifts”. The present version of the paper was circulated as a UCLA preprint since November 2001.

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Supported in part by NSF Grants 9801324 and 0100883.