Generalizations of Weakly Peripherally Multiplicative Maps Between Uniform Algebras
نویسندگان
چکیده
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 σπ(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 σπ(T (f)T (g)) ∩ σπ(fg) 6= ∅ 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 T1, T2 : A → B are surjective mappings that satisfy σπ(T1(f)T2(g)) ∩ σπ(fg) 6= ∅ for all f, g ∈ A, then T1(f)T2(1) = T1(1)T2(f) for all f ∈ A, and the map f 7→ T1(f)T2(1) is an isometric algebra isomorphism.
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