New characterizations of maximal ideals in algebras of continuous vector-valued functions

TL;DR

This paper provides new characterizations of maximal ideals in algebras of continuous vector-valued functions and offers a simplified proof of Hausner's classical result.

Contribution

It introduces novel characterizations of maximal ideals in C(X,A) and presents a different, more concise proof of Hausner's theorem.

Findings

New characterizations of maximal ideals in C(X,A)

Simplified proof of Hausner's theorem

Enhanced understanding of the structure of vector-valued function algebras

Abstract

Let X be a compact Hausdorf space, let A be a commutative unital Banach algebra, and let C(X,A) denote the algebra of continuous A-valued functions on $X$ equipped with the uniform norm ||f||=sup{||f(x)||:x\in X} for all f in C(X,A). Hausner, in [Proc. Amer. Math. Soc. 8(1957), 246--249], proved that M is a maximal ideal in C(X,A) if and only if there exist a point x in X and a maximal ideal N in A such that M={f in C(X,A) : f(x) in N}. In this note, we give new characterizations of maximal ideals in C(X,A). We also present a short proof of Hausner's result by a different approach.