On ergodic embeddings of factors

TL;DR

This paper explores the concept of ergodic embeddings of von Neumann factors, establishing conditions under which a continuous factor contains an ergodic hyperfinite II$_1$ subfactor, thus linking ergodicity with factor structure.

Contribution

It proves that continuous factors containing a maximal abelian subalgebra admit an ergodic embedding of the hyperfinite II$_1$ factor, connecting ergodic properties with factor structure.

Findings

Continuous factors contain ergodic copies of R.

Ergodic embeddings relate to maximal abelian subalgebras.

Conditions for ergodicity in von Neumann factors are established.

Abstract

An inclusion of von Neumann factors $M \subset \Cal M$ is {\it ergodic} if it satisfies the irreducibility condition $M'\cap \Cal M=\Bbb C$. We investigate the relation between this and several stronger ergodicity properties, such as $R$-{\it ergodicity}, which requires $M$ to admit an embedding of the hyperfinite II$_1$ factor $R\hookrightarrow M$ that's ergodic in $\Cal M$. We prove that if $M$ is {\it continuous} (i.e., non type I) and contains a maximal abelian $^*$-subalgebra of $\Cal M$, then $M\subset \Cal M$ is $R$-ergodic. This shows in particular that any continuous factor contains an ergodic copy of $R$.