Single elements and module isomorphisms of some operator algebra modules

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In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the AlgL-module U is initiated, where £ is a completely distributive subspace lattice on a Hilbert space H. Furthermore, as an application of single elements, we study module isomorphisms between norm closed AlgN-modules, where N is a nest, and obtain the following result: Suppose that U, V are norm closed AlgN-modules and that Φ: U → V is a module isomorphism. Then U = V and there exists a non-zero complex number A such that Φ(T) = λT, VT ∈ U.