Multiplicative spectral functions on some Banach function algebras

Abstract

In this paper, we study multiplicative functions $\phi :A\to \mathbb{C}$ on a natural Banach function algebra $A$ on a compact Hausdorff space $X$, such that $\phi (f)\in \sigma (f)$ for all $f\in A$. It is shown that for certain natural Banach function algebras $A$, either $\ker (\phi )$ is a maximal ideal of $A$ or $1\in \mathrm{span}(\ker (\phi ))$ (that is $1=f_1+f_2+\cdots f_n$ for some $f_1,...,f_n\in \ker (\phi )$). Then we investigate for the linearity of $\phi$ in either of cases that $\phi$ is continuous or $1\notin \mathrm{span}(\ker (\phi )$. 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 $\phi (f)=f(x_0)$ 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.