On ultraproduct embeddings and amenability for tracial von Neumann algebras

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 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 Connes Embedding Problem 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 $\text{Out}(N\otimes N)$ on $\mathbb{H}$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.

Updated and mildly revised. Results on commuting embeddings in both von Neumann algebraic and group theoretic contexts have been added. 27 pages. Submitted version