Multiplicatively spectrum-preserving maps of function algebras

Let $X$ be a compact Hausdorff space and $\mathcal{A}\subset C(X)$ a function algebra. Assume that $X$ is the maximal ideal space of $\mathcal A$. Denoting by $\sigma (f)$ the spectrum of an $f\in \mathcal{A}$, which in this case coincides with the range of $f$, a result of Molnár is generalized by our Main Theorem: If $\Phi :\mathcal{A} \rightarrow \mathcal{A}$ is a surjective map with the property $\sigma (fg)=\sigma (\Phi (f)\Phi (g))$ for every pair of functions $f,g\in \mathcal{A}$, then there exists a homeomorphism $\Lambda :X\rightarrow X$ such that

for every $x\in X$ and every $f\in \mathcal{A}$ with $\tau =\Phi (1)$.