Masas and Bimodule Decompositions of $\rm{II}_{1}$ Factors

TL;DR

This paper investigates the measure class in the measure-multiplicity-invariant for masas in $ m{II}_{1}$ factors, characterizing their types via bimodule analysis and providing new proofs of existing results.

Contribution

It offers a detailed study of the measure class in the measure-multiplicity-invariant and characterizes masas types using Baire category methods, along with new proofs of known theorems.

Findings

Characterization of masa types via Baire category methods

A second proof of Chifan's result on normalisers

A measure-theoretic proof linking WAHP and singularity

Abstract

The measure-multiplicity-invariant for masas in $\rm{II}_{1}$ factors was introduced in \cite{MR2261688} to distinguish masas that have the same Puk\'{a}nszky invariant. In this paper we study the measure class in the measure-multiplicity-invariant. This is equivalent to studying the standard Hilbert space as an associated bimodule. We characterize the type of any masa depending on the left-right-measure using Baire category methods (selection principle of Jankov and von Neumann). We present a second proof of Chifan's result on normalisers and a measure theoretic proof of the equivalence of weak asymptotic homomorphism property (WAHP) and singularity that appeared in \cite{MR2417416}.