Existential closedeness and the structure of bimodules of II₁ factors

2023/06/01 by Adrian Ioana, Hui Tan, Ioana, Adrian +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Logic (math.LO) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2306.00474

openalex publication_date 2023/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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↦∑_i=1^ka_i^*xa_i, for some k∈\mathbb N and (a_i)_i=1^k⊂ 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.

Related

• Existentially closed II₁ factors
• Coarse decomposition of II₁ factors
• A Note on Bimodules and II₁-Subfactors
• Bimodules in crossed products and regular inclusions of finite factors
• Tensor product indecomposability results for existentially closed factors