Strong Singularity of Singular Masas in II_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 $A=wA{w}^{\ast }$ must lie in $A$. A strongly singular masa $A$ is one that satisfies the inequality $$\Vert E_A-E_{wA{w}^{\ast }}{\Vert }_{\infty ,2}\ge \Vert w-E_A(w){\Vert }_2$$ for all unitaries $w\in N$, where $E_A$ is the conditional expectation of $N$ onto $A$, and $\Vert \cdot {\Vert }_{\infty ,2}$ is defined for bounded maps $\varphi :N\to N$ by $\sup \{\Vert \varphi (x){\Vert }_2:x\in N,\Vert x\Vert \le 1\}$. Strong singularity easily implies singularity, and the main result of this paper shows the reverse implication.

10 pages

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