THE BSE-PROPERTIES FOR VECTOR-VALUED Lp-ALGEBRAS

Abstract

Let $\mathcal{𝒜}$ be a separable Banach algebra, $G$ be a locally compact group and . We first provide a necessary and sufficient condition for which $L^p(G,\mathcal{𝒜})$ is a Banach algebra, under convolution product. Then we characterize the character space of $L^p(G,\mathcal{𝒜})$, in the case where $\mathcal{𝒜}$ is commutative and $G$ is abelian. Moreover, we investigate the BSE-property for $L^p(G,\mathcal{𝒜})$ and prove that $L^p(G,\mathcal{𝒜})$ is a BSE-algebra if and only if $\mathcal{𝒜}$ is a BSE-algebra and $G$ is finite. Finally, we study the BSE-norm property for $L^p(G,\mathcal{𝒜})$ and show that if $L^p(G,\mathcal{𝒜})$ is a BSE-norm algebra then $\mathcal{𝒜}$ is so. We prove the converse of this statement for the case where $G$ is finite and $\mathcal{𝒜}$ is a unital BSE-algebra.