Existential closedness and the structure of bimodules of II1 factors

We prove that if a separable II1 factor M is existentially closed, then every M-bimodule is weakly contained in the trivial M-bimodule, L2(M), and, equivalently, every normal completely positive map on M is a pointwise 2-norm limit of maps of the form x -> sigma ki=1 a*i xai, for some k E N and (ai)ki=1 c M. This provides the first examples of non-hyperfinite separable II1 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 II1 factors. While existentially closed II1 factors have property Gamma, by adapting this proof we construct non-Gamma II1 factors which are existentially closed in every weakly coarse extension.(c) 2023 Elsevier Inc. All rights reserved.