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

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 ${\mathcal A}\times_\phi {\mathcal B}$, the $\phi$-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 ${\mathcal A}\times_\phi {\mathcal B}$ is a BSE-algebra if and only if both ${\mathcal A}$ and ${\mathcal B}$ are BSE.