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VOL. 46 | 2017 Mixing and Spectral Gap relative to Pinsker Factors for Sofic Groups

Abstract

Motivated by our previous results, we investigate structural properties of probability measurepreserving 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 $/Gamma$.

Information

Published: 1 January 2017
First available in Project Euclid: 21 February 2017

zbMATH: 06990155
MathSciNet: MR3635672

Rights: Copyright © 2017, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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