Journal of Operator Theory
Volume 72, Issue 1, Summer 2014 pp. 87-114.
Non-selfadjoint double commutant theorems
Authors:
L.W. Marcoux (1) and M. Mastnak (2)
Author institution: (1) Department of Pure Mathematics,
University of Waterloo,
Waterloo, Ontario,
Canada N2L 3G1
(2) Department of Mathematics and Computing Science,
Saint Mary's University,
Halifax, Nova Scotia,
Canada B3H 3C3
Summary: The von Neumann double commutant theorem states that
if $\mathcal{N}$ is a
weak-operator topology closed unital,
selfadjoint subalgebra of the set $\mathcal{B}(\mathcal{H})$ of all bounded linear operators
acting on a Hilbert space $\mathcal{H}$, and if $\mathcal{N}^{\prime}
:= \{ T \in \bofh: T N
= N T \mbox{ for all } N \in \mathcal{N}\}$ denotes the commutant of
$\mathcal{N}$, then
$\mathcal{N}^{\prime \prime} = \mathcal{N}$. In this paper, we continue the
analysis of not
necessarily selfadjoint subalgebras $\mathcal{S}$ of
$\mathcal{B}(\mathcal{H})$ whose second commutant
$\mathcal{S}^{\prime \prime}$ agrees with $\mathcal{S}$.
More specifically, we examine the
case where $\mathcal{S} = \mathcal{D}+ \mathcal{R}$, where $\mathcal{R}$ is a
bimodule over a masa $\mathcal{M}$ in
$\mathcal{B}(\mathcal{H})$ and $\mathcal{D}$ is a unital subalgebra of
$\mathcal{M}$.
DOI: http://dx.doi.org/10.7900/jot.2012nov02.1968
Keywords: relative double commutant, double commutant property,
masa, skeleton, non-selfadjoint bimodule, relative annihilator
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