On ergodic embeddings of factors
arXiv:1910.06923 · doi:10.1007/s00220-020-03865-3
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 -{\it ergodicity}, which requires to admit an embedding of the hyperfinite II factor that's ergodic in $\Cal M$. We prove that if is {\it continuous} (i.e., non type I) and contains a maximal abelian -subalgebra of $\Cal M$, then $M\subset \Cal M$ is -ergodic. This shows in particular that any continuous factor contains an ergodic copy of .
July 2020: Updated to take into account the recent Das-Peterson double-ergodicity theorem for II1 factors (see 2nd part of Theorem 1.1 and comments around Problem 7.4). Paper dedicated to the memory of Dick Kadison, to appear in Communications Math Physics