The multiplier algebra and BSE-functions for certain product of Banach algebras

Abstract

In this paper, we characterize the (left) multiplier algebra of a semidirect product algebra A = B ⊕I, where I and B are closed two-sided ideal and closed subalgebra of A, respectively. As an application of this result we investigate the BSE-property of this class of Banach algebras. We then for two commutative semisimple Banach algebras A and B characterize the BSE-functions on the carrier space of A × φ B, the φ-Lau product of A and B, in terms of those functions on carrier spaces of A and B. We also prove that A × φ B is a BSE-algebra if and only if both A and B are BSE.