| Let A ⊂ C(X) and B ⊂ C(Y) be uniform algebras with Choquet boundaries delta A and deltaB. We establish sufficient conditions for a surjective map T : A → B to be an algebra isomorphism. In particular, we show that if T : A → B is a surjection that preserves the norm of the sums of the moduli of algebra elements, then T induces a homoemorphism psi between the Choquet boundaries of A and B such that |T f| = |f ∘ psi| on the Choquet boundary of B. If, in addition, T preserves the norms of all linear combinations of algebra elements and either preserves both 1 and i or the peripheral spectra of C -peaking functions, then T is a composition operator and thus an algebra isomorphism. We also show that if a surjection T that preserves the norm of the sums of the moduli of algebra elements also preserves the norms of sums of algebra elements as well as either preserving both 1 and i or preserving the peripheral spectra of C -peaking functions, then T is a composition operator and thus an algebra isomorphism. In the process, we generalize the additive analog of Bishop's Lemma. |
