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Shilov boundary

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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.

Contents

Precise definition and existence

Let A be a commutative Banach algebra and let Δ A be its structure space equipped with the relative weak*-topology of the dual A . A closed (in this topology) subset F of Δ A is called a boundary of A if max f Δ A | x ( f ) | = max f F | x ( f ) | for all x A . The set S = { F : F  is a boundary of  A } is called the Shilov boundary. It has been proved by Shilov that S is a boundary of A .

Thus one may also say that Shilov boundary is the unique set S Δ A which satisfies

  1. S is a boundary of A , and
  2. whenever F is a boundary of A , then S F .

Examples

  • Let D = { z C : | z | < 1 } be the open unit disc in the complex plane and let
  • A = H ( D ) C ( D ¯ ) be the disc algebra, i.e. the functions holomorphic in D and continuous in the closure of D with supremum norm and usual algebraic operations. Then Δ A = D ¯ and S = { | z | = 1 } .

    References

    Shilov boundary Wikipedia