Homage \`{a} A. M. Gleason et I. J. Schark

TL;DR

This paper investigates the maximal ideal space of the algebra of bounded holomorphic functions on the unit ball in complex n-space, analyzing its structure through Gleason parts, fibers, and boundary concepts using advanced function theory.

Contribution

It extends the analysis of the maximal ideal space of $H^ abla(B)$ by incorporating inner functions and boundary structures, building on prior foundational work.

Findings

Detailed characterization of Gleason parts and fibers.

Insights into the Svlov boundary structure.

Application of Aleksandrov and Hakim/Lw/Sibony's inner functions construction.

Abstract

We study the maximal ideal space of $H^\infty(B)$, where $B$ is the unit ball of $\CC^n$. Following the lead of Gleason and Schark, we analyze Gleason parts, fibers, the S\u{\i}lov boundary, and other aspects of this Banach algebra. Our work here makes good use of the inner functions construction of Aleksandrov and Hakim/L{\o} w/Sibony, particularly as formulated by Rudin.