Gleason parts for algebras of holomorphic functions in infinite dimensions

Abstract

For a complex Banach space X with open unit ball \(B_X,\) consider the Banach algebras \(\mathcal {H}^\infty (B_X)\) of bounded scalar-valued holomorphic functions and the subalgebra \(\mathcal {A}_u(B_X)\) of uniformly continuous functions on \(B_X.\) Denoting either algebra by \(\mathcal {A},\) we study the Gleason parts of the set of scalar-valued homomorphisms \(\mathcal {M}(\mathcal {A})\) on \(\mathcal {A}.\) Following remarks on the general situation, we focus on the case \(X = c_0,\) giving a complete characterization of the Gleason parts of \(\mathcal {M}(\mathcal {A}_u(B_{c_0}))\) and, among other things, showing that every fiber in \(\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))\) over a point in \(B_{\ell _\infty }\) contains \(2^c\) discs lying in different Gleason parts.