Vector-valued characters on Vector-valued Function Algebras

TL;DR

This paper introduces and studies $A$-characters on $A$-valued function algebras, showing they generalize evaluation homomorphisms and are key to understanding vector-valued spectra, especially in natural algebra cases.

Contribution

It defines $A$-characters on $A$-valued function algebras and proves their uniqueness in natural cases, extending the concept of characters to vector-valued functions.

Findings

$A$-characters generalize evaluation homomorphisms.

In natural $A$-valued algebras, $A$-characters are uniquely determined by points in $X$.

Vector-valued characters help identify vector-valued spectra.

Abstract

Let $A$ be a commutative Banach algebra and $X$ be a compact space. The class of Banach $A$-valued function algebras on $X$ consists of subalgebras of $C(X,A)$ with certain properties. We introduce the notion of $A$-characters on an $A$-valued function algebra $\A$ as homomorphisms from $\A$ into $A$ that basically have the same properties as the evaluation homomorphisms $\cE_x:f\mapsto f(x)$, with $x\in X$. For the so-called natural $A$-valued function algebras, such as $C(X,A)$ and $\Lip(X,A)$, we show that $\cE_x$ ($x\in X$) are the only $A$-characters. Vector-valued characters are utilized to identify vector-valued spectrums. When $A=\C$, Banach $A$-valued function algebras reduce to Banach function algebras, and $A$-characters reduce to characters.