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

Abstract

Suppose $M$ is a tracial von Neumann algebra embeddable into ${\mathcal{R}}^{\omega }$ (the ultraproduct of the hyperfinite $I{I}_1$-factor) and $X$ is an $n$-tuple of selfadjoint generators for $M$. Denote by $\Gamma (X;m,k,\gamma )$ the microstate space of $X$ of order $(m,k,\gamma )$. We say that $X$ is tubular if for any $\epsilon >0$ there exist $m\in \mathbb{N}$ and $\gamma >0$ such that if $(x_1,...,x_n),(y_1,...,y_n)\in \Gamma (X;m,k,\gamma ),$ then there exists a $k\times k$ unitary $u$ satisfying $|u{x}_i{u}^{\ast }-y_i{|}_2<\epsilon$ for each $1\le i\le n.$ 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 ${\mathcal{R}}^{\omega }$ are conjugate by a unitary u in ${\mathcal{R}}^{\omega }$.

6 pages, corrected typos, additional comments and references