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Amenability,tubularity, and embeddings into $$mathcal{R}^{omega}$$

Authors:

Kenley Jung

Affiliation:

(1) Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

Abstract:

Suppose M is a tracial von Neumann algebra embeddable into $mathcal{R}^{omega}$ (the ultraproduct of the hyperfinite II₁-factor) and X is an n-tuple of selfadjoint generators for M. Denote by Γ(X; m, k, γ) the microstate space of X of order (m, k ,γ). We say that X is tubular if for any ε > 0 there exist $$m in mathbb{N}$$

and γ > 0 such that if $(x_{1},ldots, x_{n}), (y_{1}, ldots, y_{n}) in Gamma(X;m,k,gamma),$ then there exists a k × k unitary u satisfying $$|ux_iu^* - y_i|_2 < epsilon$$

for each 1 ≤ i ≤ n. We show that the following conditions are equivalent:

•	M is amenable (i.e., injective).
•	X is tubular.
•	Any two embeddings of M into $mathcal{R}^{omega}$ are conjugate by a unitary $u in mathcal {R}^{omega}$ .

Research supported in part by the NSF. Dedicated to Ed Effros on the occasion of his 70th birthday.

Keywords:

KeywordHeading" >Mathematics Subject Classification (2000) Primary 46L54 Secondary 46L10

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