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

Suppose $M$ is a tracial von Neumann algebra embeddable into $\mathcal R^{\omega}$(theultraproductofthehyperfinite$II_1$-factor)and$X$isan$n$-tupleofselfadjointgeneratorsfor$M$\mathrm{.}Denoteby$\Gamma(X;m,k,\gamma)$themicrostatespaceof$X$oforder$(m,k,\gamma)$\mathrm{.}Wesaythat$X$ is tubular if for any $ϵ>0$ there exist $m\in \mathbb{N}$ and $\gamma >0$ such that if $(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n),(y_1,\mathrm{.}\mathrm{.}\mathrm{.},y_n)\in \mathrm{\Gamma }(X;m,k,\gamma ),$ then there exists a $k\times k$ unitary $u$ satisfying $\mathrm{\mid }u{x}_i{u}^{\ast }-y_i{\mathrm{\mid }}_2<ϵ$ for each $1\le i\le n\mathrm{.}$ 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 $\mathcal R^{\omega}$.