On the character space of Banach vector-valued function algebras

TL;DR

This paper explores the structure of character spaces in Banach algebra-valued function algebras, identifying conditions for their homeomorphism to product spaces and introducing a class of natural algebras.

Contribution

It introduces and studies natural A-valued function algebras, providing conditions for their character spaces to be homeomorphic to product spaces involving scalar-valued function algebras.

Findings

Character spaces of admissible A-valued function algebras can be homeomorphic to product spaces under certain conditions.

The class of natural A-valued function algebras is characterized and studied.

Examples illustrate the theoretical results.

Abstract

Given a compact space $X$ and a commutative Banach algebra $A$, the character spaces of $A$-valued function algebras on $X$ are investigated. The class of natural $A$-valued function algebras, those whose characters can be described by means of characters of $A$ and point evaluation homomorphisms, is introduced and studied. For an admissible Banach $A$-valued function algebra $\mathcal{A}$ on $X$, conditions under which the character space $M(\mathcal{A})$ is homeomorphic to $M(\mathfrak{A}) \times M(A)$ are presented, where $\mathfrak{A}=C(X) \cap \mathcal{A}$ is the subalgebra of $\mathcal{A}$ consisting of scalar-valued functions. An illustration of the results is given by some examples.