The Koopman representation and positive Rokhlin entropy

TL;DR

This paper explores the relationship between positive entropy in group actions and the Koopman representation, showing that positive entropy implies the Koopman representation embeds into the left-regular representation, with implications for mixing and ergodicity.

Contribution

It establishes a precise connection between positive entropy and Koopman representation embeddings for arbitrary countably infinite groups, extending previous results to non-amenable groups and non-free actions.

Findings

Positive entropy implies Koopman representation embeds into the left-regular representation.

Actions with completely positive outer entropy have Koopman representations isomorphic to countable sums of the left-regular representation.

Actions with completely positive outer entropy are mixing; non-amenable cases are strongly ergodic with spectral gap.

Abstract

In a prior paper, the author generalized the classical factor theorem of Sinai to actions of arbitrary countably infinite groups. In the present paper, we use this theorem and the techniques of its proof in order to study connections between positive entropy phenomena and the Koopman representation. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. By combining this result with the generalized factor theorem of the previous paper, we conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the countable direct sum of the left-regular representation. This generalizes a theorem of Dooley--Golodets for countable amenable groups. As a final consequence, we observe that actions with completely positive outer entropy must be mixing, and when the group is non-amenable they must be strongly ergodic and have spectral gap. We also prove generalizations of these results that apply relative to sub-$\sigma$-algebras and apply to non-free actions.