Regularity conditions for vector-valued function algebras

TL;DR

This paper explores how regularity conditions in vector-valued function algebras relate to their scalar and base algebra components, establishing conditions for the transfer of regularity properties.

Contribution

It provides new insights into when regularity conditions in vector-valued function algebras can be inferred from their scalar and base algebra substructures.

Findings

Regularity conditions can be transferred from scalar and base algebras to vector-valued algebras under certain conditions.

The results include applications to tensor products of commutative Banach algebras.

The paper characterizes the inheritance of regularity properties in vector-valued function algebras.

Abstract

We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin's condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra $A$ and a compact space $X$, let $\mathcal{A}$ be a Banach $A$-valued function algebra on $X$ and let $\mathfrak{A}$ be the subalgebra of $\mathcal{A}$ consisting of scalar-valued functions. This paper is about the connection between regularity conditions of the algebra $\mathcal{A}$ and the associated algebras $\mathfrak{A}$ and $A$. That $\mathcal{A}$ inherits a certain regularity condition $P$ to $\mathfrak{A}$ and $A$ is the easy part of the problem. We investigate the converse and show that, under certain conditions, $\mathcal{A}$ receives $P$ form $\mathfrak{A}$ and $A$. The results apply to tensor products of commutative Banach algebras as they are included in the class of vector-valued function algebras.