A Characterization for an L(μ, K)-Topological Module to Admit Enough Canonical Module Homomorphisms

Abstract

Let K be the scalar field of all real or complex numbers, let (Ω,A,μ) be a σ-finite measure space, and let L(μ,K) be the algebra of the μ-equivalence classes of all K-valued μ-measurable functions defined on (Ω,A,μ). L(μ,K) is a topological algebra over K when endowed with the topology of convergence locally in measure; topological modules over this topological algebra L(μ,K) (briefly, L(μ,K)-topological modules) are an extensive class of topological modules, which arise naturally in the course of the study of the theory of probabilistic normed spaces. The purpose of this paper is to show that an arbitrary regular L(μ,K)-topological module admits enough canonical module homomorphisms if and only if all of its quasi-free submodules of finite rank are complemented in the sense of topological modules.