Von Neumann Algebras and Extensions of Inverse Semigroups

TL;DR

This paper introduces a new algebraic classification of certain von Neumann algebras using inverse semigroup extensions, offering a different perspective from traditional measure-theoretic approaches.

Contribution

It provides a novel algebraic framework for classifying von Neumann algebras with Cartan MASAs via inverse semigroup extensions, and relates this to spectral theorems and subdiagonal algebras.

Findings

New classification scheme using inverse semigroup extensions

Reformulation of spectral theorem for bimodules

Description of maximal subdiagonal algebras

Abstract

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan MASAs using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper, we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman-Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.