Every countably infinite group is almost Ornstein (Q2906450)

A countable group \(G\) is called ``Ornstein'' if for every pair of probability spaces \(\left( (K,\kappa),(L,\lambda)\right)\) with the same Shannon entropy, the Bernoulli shift \(G\curvearrowright (K^G,\kappa^G)\) is isomorphic to \(G\curvearrowright (L^G,\lambda^G)\). It is well known that no finite group is Ornstein, and every countably infinite amenable group is Ornstein. NEWLINENEWLINENEWLINEIn this paper, the author introduce the concept of ``almost Ornstein''. Precisely, a group \(G\) is called ``almost Ornstein'' if whenever \((K,\kappa),(L,\lambda)\) are standard probability spaces with the same Shannon entropy, neither of which is a two-atom space, then \(G\curvearrowright (K^G,\kappa^G)\) is isomorphic to \(G\curvearrowright (L^G,\lambda^G)\). The main result is that every countably infinite group is almost Ornstein. The main ingredients of the proof of the main theorems are Thouvenot's relative isomorphism theorem for actions of \(\mathbb{Z}\), the fact that the full group of any p.m.p. aperiodic equivalence relation contains an aperiodic automorphism, and a co-induction argument similar in spirit to \textit{A. M. Stepin}'s [Sov. Math., Dokl. 16, 886--889 (1975); translation from Dokl. Akad. Nauk SSSR 223, 300--302 (1975; Zbl 0326.28026)].NEWLINENEWLINEFor the entire collection see [Zbl 1237.37004].