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$\phi$-Lauproductof${\mathcal A}$and${\mathcal B}$, in terms of those functions on carrier spaces of $\mathcal{A}$ and ${\mathcal B}$\mathrm{.}Wealsoprovethat${\mathcal A}\times_\phi {\mathcal B}$ is a BSE-algebra if and only if both $\mathcal{A}$ and $\mathcal{B}$ are BSE.