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 A as homomorphisms from A into A that basically have the same properties as evaluation homomorphisms Ex:f↦f(x), with x∈X. We show that, under certain conditions, there is a one-to-one correspondence between the set of A-characters of A and the set of characters of the function algebra A=A∩C(X) of all scalar-valued functions in A. For the so-called natural A-valued function algebras, such as C(X,A) and Lip(X,A), we show that Ex (x∈X) are the only A-characters. Vector-valued characters are utilized to identify vector-valued spectra.