Strong Singularity of Singular Masas in II_1 Factors

Details Strong Singularity of Singular Masas in II_1 Factors Strong Singularity of Singular Masas in II_1 Factors

2006-01-24

arxiv.org/abs/math/0601594v1

Illinois J. Math. 51 (4) 2007, 1077-1084

Mathematics

Operator Algebras

10 pages

Scientific paper

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.

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