Sofic Entropy of Gaussian Actions

TL;DR

This paper computes the sofic entropy of Gaussian actions associated with orthogonal representations of countable groups, extending previous results beyond abelian groups using advanced harmonic analysis and $C^{*}$-algebra techniques.

Contribution

It introduces a method to calculate the sofic entropy of Gaussian actions for general groups, not limited to abelian or amenable cases, using noncommutative harmonic analysis.

Findings

Entropy formulas for Gaussian actions of sofic groups derived

Results extend known entropy computations beyond abelian groups

New techniques involving $C^{*}$-algebras developed

Abstract

Associated to any orthogonal representation of a countable discrete group is an probability measure-preserving action called the Gaussian action. Using the Polish model formalism we developed before, we compute the entropy (in the sense of Bowen, Kerr-Li) of Gaussian actions when the group is sofic. Computations of entropy for Gaussian actions has only been done when the acting group is abelian and thus our results are new even in the amenable case. Fundamental to our approach are methods of noncommutative harmonic analysis and $C^{*}$-algebras which replace the Fourier analysis used in the abelian case.