The BSE-property for vector-valued $L^p-$algebras

TL;DR

This paper characterizes when vector-valued $L^p$-algebras are Banach algebras, explores their character spaces, and establishes conditions under which they possess the BSE-property, linking properties of the algebra and the underlying group.

Contribution

It provides necessary and sufficient conditions for $L^p(G, \\mathcal{A})$ to be a Banach algebra and characterizes the BSE-property in relation to the algebra and group structure.

Findings

$L^p(G, \\mathcal{A})$ is a Banach algebra under specific conditions.

The character space of $L^p(G, \\mathcal{A})$ is characterized for commutative $\\mathcal{A}$ and abelian $G$.

$L^p(G, \\mathcal{A})$ is a BSE-algebra iff $\\mathcal{A}$ is BSE and $G$ is finite.

Abstract

Let $\mathcal A$ be a separable Banach algebra, $G$ be a locally compact Hausdorff group and $1< p<\infty$. In this paper, 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 unital.