BSE Banach Modules and Multipliers

We introduce a class of Banach modules which are a module version of the BSE commutative Banach algebras considered by the author and Hatori. A Banach module which belongs to this class is called a BSE. Every multiplier of a BSE Banach A-module can be characterized as a continuous vector field on the carrier space of A which satisfies a Bochner-Schoenberg-Eberlein-type inequality. We give typical BSE Banach modules with respect to C*-algebras and convolution algebras. We also show that any Banach A-module is BSE whenever A is a commutative C*-algebra with discrete carrier space.