Generalizations of weakly peripherally multiplicative maps between uniform algebras

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For f ∈ A , the peripheral spectrum of f, denoted by σ π ( f ) = { λ ∈ σ ( f ) : | λ | = ‖ f ‖ } , is the set of spectral values of maximum modulus. A map T : A → B is weakly peripherally multiplicative if σ π ( T ( f ) T ( g ) ) ∩ σ π ( f g ) ≠ ∅ for all f , g ∈ A . We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T 1 , T 2 : A → B are surjective mappings that satisfy σ π ( T 1 ( f ) T 2 ( g ) ) ∩ σ π ( f g ) ≠ ∅ for all f , g ∈ A , then T 1 ( f ) T 2 ( 1 ) = T 1 ( 1 ) T 2 ( f ) for all f ∈ A , and the map f ↦ T 1 ( f ) T 2 ( 1 ) is an isometric algebra isomorphism.