Multiplicative spectral functions on some Banach function algebras

TL;DR

This paper characterizes multiplicative spectral functions on certain Banach function algebras, showing they are often linear and correspond to evaluation at a point, thus acting as characters.

Contribution

It establishes conditions under which multiplicative spectral functions are linear and identifies them as point evaluations in various Banach algebras.

Findings

Spectral functions are either associated with maximal ideals or span the identity.

Under certain conditions, spectral functions are linear and continuous.

Such functions are shown to be point evaluations, i.e., characters.

Abstract

In this paper, we study multiplicative functions $\varphi \colon A \to \Bbb C$ on a natural Banach function algebra $A$ on a compact Hausdorff space $X$, such that $\varphi(f)\in \sigma(f)$ for all $f\in A$. It is shown that for certain natural Banach function algebras $A$, either $\ker(\varphi)$ is a maximal ideal of $A$ or $1\in {\rm span}({\rm ker}(\varphi))$ (that is $1=f_1+f_2+\cdots f_n$ for some $f_1,..., f_n \in {\rm ker}(\varphi)$). Then we investigate for the linearity of $\varphi$ in either of cases that $\varphi$ is continuous or $1\notin {\rm span}({\rm ker}(\varphi)$. We show that, for some natural Banach function algebras $A$, in either of these cases, there exists a point $x_0\in X$ such that $\varphi(f)=f(x_0)$ for some family of functions $f\in A$ (including those functions $f\in A$ that $\overline{f}\in A$). In particular, such a multiplicative spectral function on some Banach algebras including $C(X)$, Lipschitz algebras, Banach algebras of absolutely continuous functions on $[0,1]$ and $C^1([0,1])$ is linear and hence it is a character.