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 $\mathcal{A}=\mathcal{B}\oplus \mathcal{I}$, where $\mathcal{I}$ and $\mathcal{B}$ are closed two-sided ideal and closed subalgebra of $\mathcal{A}$, 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 $\mathcal{A}$ and $\mathcal{B}$ characterize the BSE-functions on the carrier space of $\mathcal{A}{\times }_{\varphi }\mathcal{B}$, the $\varphi$-Lau product of $\mathcal{A}$ and $\mathcal{B}$, in terms of those functions on carrier spaces of $\mathcal{A}$ and $\mathcal{B}$. We also prove that $\mathcal{A}{\times }_{\varphi }\mathcal{B}$ is a BSE-algebra if and only if both $\mathcal{A}$ and $\mathcal{B}$ are BSE.

Cited by in corpus (1)

• Triviality of the generalized Lau product associated to a Banach algebra homomorphism