$L_{1}$- Properties of vector-valued Banach algebras

TL;DR

This paper investigates the properties of vector-valued Banach algebras associated with locally compact groups, establishing conditions under which certain measure algebras coincide and exploring their convolution structures.

Contribution

It characterizes when the measure algebra $M(G, A)$ equals the integrable algebra $L^{1}(G, A)$, and examines properties of vector-valued measure algebras on groups.

Findings

$M(G, A) = L^{1}(G, A)$ if and only if $G$ is discrete.

Properties of vector-valued measure algebras are established.

$M(G, A)$ can be a convolution measure algebra.

Abstract

Let $G$ be a locally compact group and $A$ be a commutative semisimple Banach algebra over the scalar field $\mathbb{C}$. The correlation between different types of $BSE$- Banach algebras $A$, and the Banach algebras $L^{1}(G, A)$ are assessed. It is found and approved that $M(G, A) = L^{1}(G, A)$ if and only if $G$ is discrete. Furthermore, some properties of vector-valued measure algebras on groups are given, so that $M(G, A)$ is a convolution measure algebra.