Vector-valued characters on vector-valued function algebras

Abstract

Let $A$ be a commutative unital Banach algebra and let $X$ be a compact space. We study the class of $A$-valued function algebras on $X$ as subalgebras of $C(X,A)$ with certain properties. We introduce the notion of $A$-characters of an $A$-valued function algebra $\mathcal{A}$ as homomorphisms from $\mathcal{A}$ into $A$ that basically have the same properties as evaluation homomorphisms ${\mathcal{E}}_x:f\mapsto f\left(x\right)$, with $x\in X$. We show that, under certain conditions, there is a one-to-one correspondence between the set of $A$-characters of $\mathcal{A}$ and the set of characters of the function algebra $\mathfrak{A}=\mathcal{A}\cap C\left(X\right)$ of all scalar-valued functions in $\mathcal{A}$. For the so-called natural $A$-valued function algebras, such as $C(X,A)$ and $Lip(X,A)$, we show that ${\mathcal{E}}_x$ ($x\in X$) are the only $A$-characters. Vector-valued characters are utilized to identify vector-valued spectra.