Characterizations of Double Commutant Property on B H

Abstract

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