Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On ultraproduct embeddings and amenability for tracial von Neumann
  algebras   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.

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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: econ