Strong singularity of singular masas in ${\rm II}\sb 1$ factors

Abstract

A singular masa $A$ in a ${\mathrm{II}}_1$ 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

\[ \|\bb E_A-\bb E_{wAw^*}\|_{\infty,2}\geq\|w-\bb E_A(w)\|_2 \]

for all unitaries $w\in N$, where $\bb E_A$ 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.