2023/01/27 by Gábor Szabó, Szabó, Gábor, Lise Wouters +1
Mathematics · #46L55 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Geometric and Algebraic Topology #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2301.11748
openalex publication_date 2023/01/27 · openalex created_date 2023/01/31 · openalex updated_date 2026/07/28
We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II1-factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C^*-dynamics. Given a countable discrete group G and an amenable action G\curvearrowright M on any separably acting semi-finite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing G-action is suitably absorbed at the level of each fibre in the direct integral decomposition of M, then it is tensorially absorbed by the action on M. As a direct application of Ocneanu's theorem, we deduce that if M has the McDuff property, then every amenable G-action on M has the equivariant McDuff property, regardless whether M is assumed to be injective or not. By employing Tomita-Takesaki theory, we can extend the latter result to the general case where M is not assumed to be semi-finite.