On Ultraproduct Embeddings and Amenability for Tracial von Neumann Algebras. (26th October 2020)

• Record Type: Journal Article
• Authors: Atkinson, Scott; Kunnawalkam Elayavalli, Srivatsav
• Is Part Of: International mathematics research notices. Volume 2021:Number 4(2021)
• Page Start: 2882
• Page End: 2918
• Publication Date: 2020-10-26
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa257
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes embedding problem (CEP) is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the CEP is amenable if and only if any two embeddings into $R^{\mathcal{U}}$ are ucp-conjugate. Moreover, we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$ om$(N, \prod _{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of embeddings is separable if and only $N$ is hyperfinite. This resolves a question of Popa for Connes embeddable factors. These results hold when we further ask that the pairs of embeddings commute, admitting a nontrivial action of $\textrm{Out}(N\otimes N)$ on ${\mathbb{H}}\textrm{om}(N\otimes N, \prod _{k\to \mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous result for commuting sofic representations of countable sofic groups.