Existential closedeness and the structure of bimodules of II$_1$ factors

Abstract

We prove that if a separable II$_1$ factor $M$ is existentially closed, then every $M$-bimodule is weakly contained in the trivial $M$-bimodule, $\text{L}^2(M)$, and, equivalently, every normal completely positive map on $M$ is a pointwise 2-norm limit of maps of the form $x\mapsto\sum_{i=1}^ka_i^*xa_i$, for some $k\in\mathbb N$ and $(a_i)_{i=1}^k\subset M$. This provides the first examples of non-hyperfinite separable II$_1$ factors $M$ with the latter properties. We also obtain new characterizations of $M$-bimodules which are weakly contained in the trivial or coarse $M$-bimodule and of relative amenability inside $M$. Additionally, we give an operator algebraic presentation of the proof of the existence of existentially closed II$_1$ factors. While existentially closed II$_1$ factors have property Gamma, by adapting this proof we construct non-Gamma II$_1$ factors which are existentially closed in every weakly coarse extension.