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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 $\mathcal{M}$.

Keywords: relative double commutant, double commutant property, masa, skeleton, non-selfadjoint bimodule, relative annihilator

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