Gleason parts for algebras of holomorphic functions in infinite dimensions

Abstract

For a complex Banach space X with open unit ball BX, consider the Banach algebras H∞(BX) of bounded scalar-valued holomorphic functions and the subalgebra Au(BX) of uniformly continuous functions on BX. Denoting either algebra by A, we study the Gleason parts of the set of scalar-valued homomorphisms M(A) on A. Following remarks on the general situation, we focus on the case X= c, giving a complete characterization of the Gleason parts of M(Au(Bc0)) and, among other things, showing that every fiber in M(H∞(Bc0)) over a point in Bℓ∞ contains 2 c discs lying in different Gleason parts.