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 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
- S is a boundary of A , and
- whenever F is a boundary of A , then S ⊂ F .
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 } .
