Strong Singularity of Singular Masas in II_1 Factors

TL;DR

This paper proves that in II_1 factors, strong singularity of a masa is equivalent to singularity, establishing a key characterization of these subalgebras with implications for their structure.

Contribution

The paper demonstrates that strong singularity and singularity are equivalent properties for masas in II_1 factors, clarifying their relationship.

Findings

Strong singularity implies singularity in II_1 factors.

The main result shows the reverse: singularity implies strong singularity.

Provides a new characterization of masas in operator algebras.

Abstract

A singular masa $A$ in a $\rm{II}_{1}$ factor $N$ is defined by the property that any unitary $w\in N$ for which $A=wAw^*$ must lie in $A$. A strongly singular masa $A$ is one that satisfies the inequality $$\| E_A- E_{wAw^*}\|_{\infty,2}\geq\|w- E_A(w)\|_2$$ for all unitaries $w\in N$, where $E_A$ is the conditional expectation of $N$ onto $A$, and $\|\cdot\|_{\infty,2}$ is defined for bounded maps $\phi :N\to N$ by $\sup\{\|\phi(x)\|_2:x\in N, \|x\|\leq 1\}$. Strong singularity easily implies singularity, and the main result of this paper shows the reverse implication.