A Note on Relatively Injective \(C_0(S)\)-Modules \(C_0(S)\)

Abstract

In this note we discuss some necessary and some sufficient conditions for the relative injectivity of the \(C_0(S)\)-module \(C_0(S)\), where \(S\) is a locally compact Hausdorff space. We also give a Banach module version of Sobczyk’s theorem. The main result of the paper is as follows: if the \(C_0(S)\)-module \(C_0(S)\) is relatively injective, then \(S=\beta(S\setminus \{s\})\) for any limit point \(s\in S\).