Uniform algebra isomorphisms and peripheral multiplicativity

Let $\varphi \colon A\to B$ be a surjective operator between two uniform algebras with $\varphi (1)=1$. We show that if $\varphi$ satisfies the peripheral multiplicativity condition $\sigma _\pi \big (\varphi (f) \varphi (g)\big )=\sigma _\pi (fg)$ for all $f,g\in A$, where $\sigma _\pi (f)$ is the peripheral spectrum of $f$, then $\varphi$ is an isometric algebra isomorphism from $A$ onto $B$. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.