On ergodic embeddings of factors

An inclusion of von Neumann factors $M\subset \text{Cal}M$ is {\it ergodic} if it satisfies the irreducibility condition $M^{\mathrm{\prime }}\cap \text{Cal}M=\mathbb{C}$. 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 $\text{Cal}M$. We prove that if $M$ is {\it continuous} (i.e., non type I) and contains a maximal abelian ${}^{\ast }$-subalgebra of $\text{Cal}M$, then $M\subset \text{Cal}M$ is $R$-ergodic. This shows in particular that any continuous factor contains an ergodic copy of $R$.