φ-Multipliers on Banach Algebras and Topological Modules

Abstract

We prove some results concerning Arens regularity and amenability of the Banach algebra $M_{\phi }\left(A\right)$ of all $\phi$-multipliers on a given Banach algebra $A$. We also consider $\phi$-multipliers in the general topological module setting and investigate some of their properties. We discuss the $\phi$-strict and $\phi$-uniform topologies on $M_{\phi }\left(A\right)$. A characterization of $\phi$-multipliers on $L_1\left(G\right)$-module $L_p\left(G\right)$, where $G$ is a compact group, is given.