Illinois Journal of Mathematics

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.

Article information

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

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138533

Digital Object Identifier
doi:10.1215/ijm/1258138533

Mathematical Reviews number (MathSciNet)
MR2417416

Zentralblatt MATH identifier
1152.46050

Subjects
Primary: 46L10: General theory of von Neumann algebras
Secondary: 46L35: Classifications of $C^*$-algebras

Citation

Sinclair, Allan M.; Smith, Roger R.; White, Stuart A.; Wiggins, Alan. Strong singularity of singular masas in ${\rm II}\sb 1$ factors. Illinois J. Math. 51 (2007), no. 4, 1077--1084. doi:10.1215/ijm/1258138533. https://projecteuclid.org/euclid.ijm/1258138533