Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties

Let $T$ be a surjective mapping from a uniform algebra $A$ on a compact Hausdorff space $X$ onto a uniform algebra on a compact Hausdorff space $Y$. Suppose that $fg(X)=T(f)T(g)(Y)$ holds for every $f,g\in A$. Then we have that $T$ is an almost isometric isomorphism, which is a generalization of results of Molnár (2002) and Rao and Roy (2005).