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arXiv · 1910.06923

On ergodic embeddings of factors

Sorin Popa
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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 $R$.

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Sorin Popa. 2019-10-15. On ergodic embeddings of factors. https://doi.org/10.1007/s00220-020-03865-3

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