Multiplicative spectral functions on some Banach function algebras

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.