Convex sets associated with von~Neumann algebras and Connes' approximate embedding problem

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.

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On central traces and groups of symmetries of order unit Banach spaces

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500016126 ◽

1978 ◽

Vol 21 (2) ◽

pp. 159-166

Author(s):

Cho-Ho Chu

Keyword(s):

Convex Sets ◽

Von Neumann Algebras ◽

The Other ◽

Order Unit ◽

Approximation Process ◽

Von Neumann ◽

Easy Consequence ◽

Finite Dimensional ◽

Module Homomorphism ◽

Group Of Symmetries

A central trace on an order-unit Banach space A(K) is a centre-valued module homomorphism invariant under the group of symmetries of A(K).The concept of central traces has been crucial in the theory of types for convex sets established in (4), (5). In von Neumann algebras, they are precisely the canonical centre-valued traces and their existence hinges on a fundamental theorem (Dixmier's approximation process) in von Neumann algebras. On the other hand, the existence of central traces in finite dimensional spaces is an easy consequence of Ryll-Nardzewski's fixed point theorem (5).

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C*-convex sets and completely bounded bimodule homomorphisms

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500000202 ◽

2000 ◽

Vol 130 (2) ◽

pp. 375-387 ◽

Cited By ~ 9

Author(s):

B. Magajna

Keyword(s):

Hilbert Space ◽

Convex Sets ◽

Von Neumann Algebras ◽

Dual Operator ◽

Von Neumann ◽

Additional Requirement ◽

Convex Subsets

If A and B are C*-algebras and X is an operator A, B-bimodule, then points of X can be separated from closed A, B-absolutely convex subsets of X by completely bounded A, B-bimodule homomorphisms from X into B(K), where K is a Hilbert space and the A, B-bimodule structure on B(K) is induced by a pair of representations π : A → B(K) and σ : B → B(K). If A and B are von Neumann algebras and X is a normal (not necessarily dual) operator A, B-bimodule, those A, B-absolutely convex subsets of X are characterized which can be separated from points of X as above, but with the additional requirement that the two representations π and σ are normal. This requires a new topology on X, which has appeared also in connection with some other questions concerning operator modules.

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The Theory of Tracial Von Neumann Algebras Does Not Have A Model Companion

Journal of Symbolic Logic ◽

10.2178/jsl.7803170 ◽

2013 ◽

Vol 78 (3) ◽

pp. 1000-1004 ◽

Cited By ~ 5

Author(s):

Isaac Goldbring ◽

Bradd Hart ◽

Thomas Sinclair

Keyword(s):

Positive Solution ◽

Von Neumann Algebras ◽

Quantifier Elimination ◽

Complete Theory ◽

Embedding Problem ◽

Model Companion ◽

Von Neumann

AbstractIn this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of II1 factors.

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