Amenability, tubularity, and embeddings into \mathcal R^ω

2005/06/06 by Jung, Kenley
#FOS: Mathematics #Operator Algebras (math.OA) #Primary 46L54 #Secondary 46L10

paper · doi:10.48550/arxiv.math/0506108

Abstract

Suppose M is a tracial von Neumann algebra embeddable into \mathcal R^ω (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 ∈ \mathbb N and γ>0 such that if (x₁,..., x_n), (y₁, ..., y_n) ∈ Γ(X;m,k,γ), then there exists a k × k unitary u satisfying |ux_iu^* - y_i|₂ < ε for each 1 ≤ i ≤ n. We show that the following conditions are equivalent: 1) M is amenable (i.e., injective). 2) X is tubular; 3) Any two embeddings of M into \mathcal R^ω are conjugate by a unitary u in \mathcal R^ω.

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