Multipliers of Modules of Continuous Vector-Valued Functions

Abstract

In 1961, Wang showed that if $A$ is the commutative $C^{\mathrm{*}}$-algebra $C_{\mathrm{0}}(X)$ with $X$ a locally compact Hausdorff space, then $M(C_{\mathrm{0}}(X))\cong C_b(X)$. Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing that $\text{H}\text{o}{\text{m}}_{C_{\mathrm{0}}(X,A)}(C_{\mathrm{0}}(X,E),C_{\mathrm{0}}(X,F))\simeq C_{s,b}(X,\text{H}\text{o}{\text{m}}_A(E,F)),$ where $E$ and $F$ are $p$-normed spaces which are also essential isometric left $A$-modules with $A$ being a certain commutative $F$-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.