POLISH MODELS AND SOFIC ENTROPY

Abstract

We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if$\unicode[STIX]{x1D6E4}$is a nonamenable group and$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.