The spectra of Banach algebras of holomorphic functions on polydisk type domains

Abstract

R.M. Aron et al. [3] presented that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra H ∞ (B c0 ). On the other hand, B.J. Cole and T.W. Gamelin [15] showed that) in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets U of a Banach space X for which H ∞ (U ) is isometrically isomorphic to H ∞ (B c0 ). We prove that there exist polydisk type domains U of any infinite dimensional Banach space X with a Schauder basis such that H ∞ (U ) is isometrically isomorphic to H ∞ (B c0 ), which also generalizes the result by Cole and Gamelin [15]. We also show that the Cluster Value Theorem is true for H ∞ (U ). As the dual space X * is not necessarily contained in H ∞ (U ) for such a domain U , we find a natural way to define fibers of the spectrum of H ∞ (U ), and study the analytic and algebraic structure of its spectrum including its Gleason parts.