Single elements and module isomorphisms of some operator algebra modules

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the 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 Alg$\mathcal{N}$-modules, where $\mathcal{N}$ is a nest, and obtain the following result: Suppose that $\mathcal{U}, \mathcal{V}$ are norm closed 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 $\Phi (T)=\lambda T, \forall T\in\mathcal{U}$.