Boundaries for Gelfand transform image of Banach algebras of holomorphic functions

Abstract

In this paper, we study boundaries for the Gelfand transform image of infinite dimensional ana-logues of the classical disk algebras. More precisely, given a certain Banach algebra A of bounded holomorphic functions on the open unit ball BX of a complex Banach space X, we show that the Shilov boundary for the Gelfand transform image of A can be explicitly described for a large class of Banach spaces. Some possible application of our result to the famous Corona theorem is also briefly discussed.