On ultraproduct embeddings and amenability for tracial von Neumann\n algebras

2019/07/07 by Scott Atkinson, Srivatsav Kunnawalkam Elayavalli, Atkinson, Scott +1 · 2 citations
Mathematics · #03C20 #46Lxx #Advanced Operator Algebra Research #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1907.03359

openalex publication_date 2019/07/07 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

We define the notion of self-tracial stability for tracial von Neumann\nalgebras and show that a tracial von Neumann algebra satisfying the Connes\nEmbedding Problem is self-tracially stable if and only if it is amenable. We\nthen generalize a result of Jung by showing that a separable tracial von\nNeumann algebra that satisfies the Connes Embedding Problem is amenable if and\nonly if any two embeddings into R^\U are ucp-conjugate. Moreover we\nshow that for a II₁ factor N satisfying CEP, the space \ℍom(N,\n\∏_{k\→ \U}M_k) of unitary equivalence classes of embeddings is\nseparable if and only N is hyperfinite. This resolves a question of Popa for\nConnes embeddable factors. These results hold when we further ask that the\npairs of embeddings commute, admitting a nontrivial action of\n\Out(N\⊗ N) on \ℍom(N\⊗ N, \∏_k\→\n\UM_k) whenever N is non-amenable. We also obtain an analogous\nresult for commuting sofic representations of countable sofic groups.\n

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