Shilov boundary


In functional analysis, a branch of mathematics, 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

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.