Entropy and the variational principle for actions of sofic groups

TL;DR

This paper develops a general operator algebra approach to sofic entropy for group actions, establishing a variational principle that links measure and topological invariants, and computes entropy for specific algebraic actions.

Contribution

It introduces a unified operator algebra framework for sofic entropy, extending previous notions and establishing the variational principle for a broader class of actions.

Findings

Established the variational principle for sofic entropy.

Computed topological entropy for principal algebraic actions.

Extended entropy concepts to residually finite groups.

Abstract

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C*-algebra.