Mixing and spectral gap relative to Pinsker factors for sofic groups.(English)Zbl 1403.37009

Morrison, Scott (ed.) et al., Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday. Canberra: Australian National University, Centre for Mathematics and its Applications. Proceedings of the Centre for Mathematics and its Applications, Australian National University 46, 193-221 (2017).
Summary: Motivated by our previous results, we investigate structural properties of probability measure-preserving actions of sofic groups relative to their Pinsker factor. We also consider the same properties relative to the Outer Pinsker factor, which is another generalization of the Pinsker factor in the nonamenable case. The Outer Pinsker factor is motivated by entropy in the presence, which fixes some of the “pathological” behavior of sofic entropy: namely increase of entropy under factor maps. We show that an arbitrary probability measure-preserving action of a sofic group is mixing relative to its Pinsker and Outer Pinsker factors and, if the group is nonamenable, it has spectral gap relative to its Pinsker and Outer Pinsker factors. Our methods are similar to those we developed in “Polish models and sofic entropy” and based on representation-theoretic techniques. One crucial difference is that instead of considering unitary representations of a group $$\Gamma$$, we must consider $$\ast$$-representations of algebraic crossed products of $$L^\infty$$ spaces by $$\Gamma$$.
For the entire collection see [Zbl 1386.46002].

MSC:

 37A25 Ergodicity, mixing, rates of mixing 22D40 Ergodic theory on groups
Full Text: