Homage à A. M. Gleason et I. J. Schark

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\uılov boundary, and other aspects of this Banach algebra. Our work here makes good use of the inner functions construction of Aleksandrov and Hakim/Lø w/Sibony, particularly as formulated by Rudin.