Multipliers on Banach Algebras and Topological Modules

We prove some results concerning Arens regularity and amenability of the Banach algebra [M.sub.[phi]] (A) 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.sub.[phi]] (A). A characterization of [phi]-multipliers on [L.sub.1](G)-module [L.sub.p](G), where G is a compact group, is given.