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In [[functional analysis]], a branch of mathematics, the '''Shilov boundary''' is the smallest   [[closed set|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 <math>\mathcal A</math> be a [[commutative]] [[Banach algebra]] and let <math>\Delta \mathcal A</math> be its [[structure space]] equipped with the [[relative topology|relative]] [[weak topology|weak*-topology]] of the [[continuous dual space|dual]] <math>{\mathcal A}^*</math>. A closed (in this topology) subset <math>F</math> of <math>\Delta {\mathcal A}</math> is called a '''boundary''' of <math>{\mathcal A}</math> if <math>\max_{f \in \Delta {\mathcal A}} |x(f)|=\max_{f \in F} |x(f)|</math> for all <math>x \in \mathcal A</math>.
The set <math>S=\bigcap\{F:F \text{ is a boundary of } {\mathcal A}\}</math> is called the '''Shilov boundary'''. It has been proved by Shilov<ref>Theorem 4.15.4 in [[Einar Hille]], [[Ralph S. Phillips]]: [http://www.ams.org/online_bks/coll31/coll31-chIV.pdf Functional analysis and semigroups]. -- AMS, Providence 1957.</ref> that <math>S</math> is a boundary of <math>{\mathcal A}</math>.

Thus one may also say that Shilov boundary is the unique set <math>S \subset \Delta \mathcal A</math> which satisfies
#<math>S</math> is a boundary of <math>\mathcal A</math>, and
#whenever <math>F</math> is a boundary of <math>\mathcal A</math>, then <math>S \subset F</math>.

== Examples ==
*Let <math>\mathbb D=\{z \in \mathbb C:|z|<1\}</math> be the [[open unit disc]] in the [[complex plane]] and let 
<math>{\mathcal A}={\mathcal H}(\mathbb D)\cap {\mathcal C}(\bar{\mathbb D})</math> be the [[disc algebra]], i.e. the functions [[holomorphic]] in <math>\mathbb D</math> and [[continuous function|continuous]] in the [[closure (topology)|closure]] of <math>\mathbb D</math> with [[supremum norm]] and usual algebraic operations. Then <math>\Delta {\mathcal A}=\bar{\mathbb D}</math> and <math>S=\{|z|=1\}</math>.

== References ==
*{{Springer|id=B/b110310|title=Bergman-Shilov boundary}}

==Notes==
{{Reflist}}

== See also ==
*[[James boundary]]

[[Category:Banach algebras]]