BSE-properties of Vector-valued group algebras

TL;DR

This paper investigates the BSE-property in vector-valued group algebras, establishing that the BSE-property of $L^1(G, \\mathcal{A})$ is equivalent to that of the algebra \\mathcal{A} itself.

Contribution

It provides a characterization of the BSE-property for vector-valued group algebras, linking it directly to the property of the underlying algebra \\mathcal{A}.

Findings

$L^1(G, \\mathcal{A})$ is a BSE algebra if and only if \\mathcal{A} is a BSE algebra.

The result applies to commutative, semisimple Banach algebras with identity.

The study extends the understanding of BSE-properties in the context of vector-valued group algebras.

Abstract

Let $ \mathcal{A} $ be a commutative and semisimple Banach algebra with identity norm one and $ G $ be an abelian locally compact Hausdorff group. In this paper, we study BSE-Property for $L^1(G,\mathcal A)$ and show that $L^1(G,\mathcal A)$ is a BSE algebra if and only if $\mathcal A$ is so.