Ring isomorphisms in norm between Banach algebras of continuous complex-valued functions

Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X)$ and $C(Y)$ denote the commutative Banach algebras of all continuous complex-valued functions on $X$ and $Y$, respectively. We study bijective maps $T$ from $C(X)$ onto $C(Y)$ which preserve the ring structure in the norm in the following sense: $$\mathrm{\parallel }T(f+g)\mathrm{\parallel }=\mathrm{\parallel }T(f)+T(g)\mathrm{\parallel },\mathrm{\parallel }T(fg)\mathrm{\parallel }=\mathrm{\parallel }T(f)T(g)\mathrm{\parallel }(f,g\in C(X))\mathrm{.}$$ Our main objective is to clarify whether such maps must necessarily be induced by homeomorphisms between the underlying spaces. Under the additional assumption that $T(\bar{f})=\overline{T(f)}$ for $f\in C(X)$, we prove that $T$ is a real-linear isometry. As a consequence, we obtain a concrete representation of such maps as weighted composition operators.