SINGLE ELEMENTS IN SOME REFLEXIVE ALGEBRA MODULES

Abstract

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in Alg$\mathcal{L}$-module $\mathcal{U}_{\phi}$ is initiated, where $\mathcal{L}$ is a completely distributive subspace lattice on a Banach space $\mathcal{X}$ and $\phi$ is an order homomorphism from $\mathcal{L}$ into $\mathcal{L}$. For a reflexive Banach space $\mathcal{X}$ and a positive integer $n$ (or $+\infty$), by virtue of the order homomorphism $\phi$ we give necessary and sufficient conditions for the existence of single elements of $\mathcal{U}_{\phi}$ of rank $n$ (or $+\infty$).