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 (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.

The Connes Embedding Problem: A guided tour

“…We say that a II 1 factor M has the Jung property if every embedding of M into its ultrapower M U is unitarily conjugate to the diagonal embedding. Atkinson and Kunnawalkam Elayavalli [5] showed that R is the only R U -embeddable factor with the Jung property. However, in [31], we showed that E, if it exists, also has the Jung property.…”

Non-embeddable II$_1$ factors resembling the hyperfinite II$_1$ factor

“…(See [3,Question 3.3.12] for an explicit mention of the former question.) 2 Our first main result is that "generically" these are the same question. To explain this, recall that a tracial von Neumann algebra M is existentially closed (or e.c.…”

“…In [3], the authors say that a separable II 1 factor M has the Jung property if and only if any embedding of M into M U is unitarily conjugate to the diagonal embedding. In [2] (see also [3,Theorem 3.1.3]), the authors show that R is the unique separable embeddable II 1 factor with the Jung property.…”

“…It follows that R is self-tracially stable. The fact that R is the unique separable embeddable self-tracially stable II 1 factor is the content of [2,Theorem 2.4].…”

“…Separability is not an issue here if one allows ultrafilters over larger index sets 2. We should mention that in [3, Theorem 3.2.5] it was shown that R is the unique separable embeddable II 1 factor with the generalized Jung property, meaning that any two embeddings of itself into its ultrapower are conjugate by some (not necessarily inner) automorphism of the ultrapower;[3, Theorem 3.3.1] shows that there are non-embeddable factors with this property 3.…”