Characterizations of Double Commutant Property on  B   H

Chaoqun Chen,Ling Wang,Cuimei Cui,Fangyan Lu,Eva A Gallardo Gutiérrez

doi:10.1155/2021/6654100

Abstract

Let H be a complex Hilbert space. Denote by B H the algebra of all bounded linear operators on H . In this paper, we investigate the non-self-adjoint subalgebras of B H of the form T + B , where B is a block-closed bimodule over a masa and T is a subalgebra of the masa. We establish a sufficient and necessary condition such that the subalgebras of the form T + B has the double commutant property in some particular cases.

Highlights

We denote by BðH Þ the algebra of all bounded linear operators on H
We analyze the settings of non-self-adjoint subalgebras of BðH Þ whose double commutant coincides with themselves
We say that such algebras satisfy the double commutant property

Summary

Introduction

We analyze the settings of non-self-adjoint subalgebras of BðH Þ whose double commutant coincides with themselves. We say that such algebras satisfy the double commutant property. Ruston [16] showed that every algebraic operator in BðH Þ has the double commutant property. Marcoux and Mastnak [12] analyzed the non-self-adjoint subalgebras of BðH Þ whose double commutant agrees with themselves; they considered the class of algebras of the form D + R in finite dimensional space, where R is a bimodule over a masa and D is a unital subalgebras of the masa. We will investigate the subalgebras of BðH Þ with the double commutant property, which extends the result in [13] extensively

Preliminaries

The Double Commutant Theorem on BðH Þ