Sofic entropy of Gaussian actions

Abstract

Associated to any orthogonal representation of a countable discrete group, is a 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 [J. Amer. Math. Soc.23(2010) 217–245], Kerr and Li [Invent. Math.186(2011) 501–558]) of Gaussian actions when the group is sofic. Computation 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 non-commutative harmonic analysis and$C^{\ast }$-algebras which replace the Fourier analysis used in the abelian case.