Single elements and module isomorphisms of some operator algebra modules

Abstract:

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the $\mbox {Alg}\mathcal {L}$-module $\mathcal {U}$ is initiated, where $\mathcal {L}$ is a completely distributive subspace lattice on a Hilbert space ${\mathcal H}$. Furthermore, as an application of single elements, we study module isomorphisms between norm closed $\mbox {Alg}\mathcal {N}$-modules, where $\mathcal {N}$ is a nest, and obtain the following result: Suppose that $\mathcal {U}, \mathcal {V}$ are norm closed $\mbox {Alg}\mathcal {N}$-modules and that $\Phi : \mathcal {U}\rightarrow \mathcal {V}$ is a module isomorphism. Then $\mathcal {U}=\mathcal {V}$ and there exists a non-zero complex number $\lambda$ such that $\Phi (T)=\lambda T, \forall T\in \mathcal {U}$.