Gleason parts and weakly compact homomorphisms between uniform Banach algebras (Q1807619)

In the paper the authors continue their study of the relationship between Gleason parts and weakly compact homomorphisms on uniform Banach algebras started in [\textit{R. Aron} and the authors, Stud. Math 123, No. 3, 235-247 (1997; Zbl 0898.46049)]. Using known results concerning the Gleason parts of a uniform Banach algebra, the authors derive several consequences about weakly compact homomorphisms between uniform algebras. These results are then applied to the algebra \(H^{\infty}(\mathbb{D})\), where \(\mathbb D\) is the unit disk in \(\mathbb C\). Several equivalent conditions are given for a homomorphism on \(H^{\infty}(\mathbb D) \) to be weakly compact, as well as necessary and sufficient conditions for a homomorphism on \(H^{\infty}(\mathbb D)\) to be a composition operator.