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 $x\mapsto {\sum }_{i=1}^k{a}_i^{\ast }x{a}_i$, 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.