Every countably infinite group is almost Ornstein

TL;DR

This paper proves that every countably infinite group exhibits the property that Bernoulli shifts over it are isomorphic whenever the base spaces have the same Shannon entropy, extending the Ornstein isomorphism theorem.

Contribution

It establishes that all countably infinite groups are almost Ornstein, generalizing the Ornstein theorem to a broader class of groups.

Findings

All countably infinite groups are almost Ornstein.

Bernoulli shifts over these groups are classified by Shannon entropy.

The result applies to a wide class of groups beyond amenable ones.

Abstract

We say that a countable discrete group $G$ is {\em almost Ornstein} if for every pair of standard non-two-atom probability spaces $(K,\kappa), (L,\lambda)$ with the same Shannon entropy, the Bernoulli shifts $G \cc (K^G,\kappa^G)$ and $G \cc (L^G,\lambda^G)$ are isomorphic.