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On Ergodic Embeddings of Factors

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Abstract

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

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Notes

  1. I am grateful to Stefaan Vaes for his help with the proof of \(1^\circ \Rightarrow 4^\circ \) in Corollary 6.7.

  2. Note that the separability of the factors \(M, \mathcal {M}\) is essential. Indeed, if M is a non-Gamma II\(_1\) factor and we let \(\mathcal {M}=M^\omega \) for some free ultrafilter \(\omega \) then \(M'\cap \mathcal {M}=\mathbb {C}\), implying \(M\subset \mathcal {M}\) is MV-ergodic, while by [P81a] M contains no MASAs of \(\mathcal {M}\), nor copies of R that are ergodic in \(\mathcal {M}\).

  3. Question: does any free cocycle \(\Gamma \)-action on R have this property?

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  1. Math.Dept., UCLA, University of California, Los Angeles, Los Angeles, CA, 90095-1555, USA

    Sorin Popa

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  1. Sorin Popa
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Correspondence to Sorin Popa.

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Communicated by Y. Kawahigashi

To Dick Kadison, in memoriam

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Supported in part by NSF Grant DMS-1700344 and the Takesaki Chair in Operator Algebras.

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Popa, S. On Ergodic Embeddings of Factors. Commun. Math. Phys. 384, 971–996 (2021). https://doi.org/10.1007/s00220-020-03865-3

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