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 $II_1$-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 $|ux_iu^* - y_i|_2 < \epsilon$ for each $1 \leq i \leq n.$ 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 $\mathcal R^{\omega}$.