A cohomological characterization of approximately finite dimensional von Neumann algebras

TL;DR

This paper generalizes Connes' cohomological characterization of approximately finite dimensional von Neumann algebras by extending the derivation concept to include 2-cocycles, providing new insights into their structure.

Contribution

It introduces a cohomological framework using 2-cocycles to characterize approximately finite dimensional von Neumann algebras, extending Connes' original derivation-based approach.

Findings

Generalization of Connes' result to 2-cocycles

Identification of a new module related to Connes' module

Characterization of AFD von Neumann algebras via cohomology

Abstract

For a von Neumann algebra M on a Hilbert space, A. Connes has constructed a module S and a derivation of M into S, such that M is approximately finite dimensional if and only if that derivation is inner. The paper contains a generalization of this result to the situation with a 2-cocycle instead. The cocycle is the obvious generalization, and the module is closely related to Connes, but isn't a dual module.