Shilov boundary


In functional analysis, the Shilov boundary is the smallest closed subset of the structure space of a commutative Banach algebra where an analog of the maximum modulus principle holds. It is named after its discoverer, Georgii Evgen'evich Shilov.

Precise definition and existence

Let be a commutative Banach algebra and let be its structure space equipped with the relative weak*-topology of the dual. A closed subset of is called a boundary of if for all.
The set is called the Shilov boundary. It has been proved by Shilov that is a boundary of.
Thus one may also say that Shilov boundary is the unique set which satisfies
  1. is a boundary of, and
  2. whenever is a boundary of, then.

Examples

Let be the open unit disc in the complex plane and let be the disc algebra, i.e. the functions holomorphic in and continuous in the closure of with supremum norm and usual algebraic operations. Then and.