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- Title
On Ultraproduct Embeddings and Amenability for Tracial von Neumann Algebras.
- Authors
Atkinson, Scott; Elayavalli, Srivatsav Kunnawalkam
- 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.
- Subjects
VON Neumann algebras; COMMUTING
- Publication
IMRN: International Mathematics Research Notices, 2021, Vol 2021, Issue 4, p2882
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnaa257