Relative entropy and the Pinsker product formula for sofic groups

Abstract

We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of local and doubly empirical convergence developed by Austin we prove that in many cases the outer Pinsker factor of a product action is the product of the outer Pinsker factors. Our results are parallel to those of Seward for Rokhlin entropy. We use these Pinsker product formulas to show that if $X$ is a compact group, and $G$ is a sofic group with $G\curvearrowright X$ by automorphisms, then the outer Pinsker factor of $G\curvearrowright (X,m_X)$ is given as a quotient by a $G$-invariant, closed, normal subgroup of $X$. We use our results to show that if $G$ is sofic and $f\in M_n(\mathbb{Z}(G))$ is invertible as a convolution operator ${\ell }^2(G{)}^{\oplus n}\to {\ell }^2(G{)}^{\oplus n},$ then the action of $G$ on the Pontryagin dual of $\mathbb{Z}(G{)}^{\oplus n}/\mathbb{Z}(G{)}^{\oplus n}f$ has completely positive measure-theoretic entropy with respect to the Haar measure.