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 [math]\displaystyle{ \mathcal A }[/math] be a commutative Banach algebra and let [math]\displaystyle{ \Delta \mathcal A }[/math] be its structure space equipped with the relative weak*-topology of the dual [math]\displaystyle{ {\mathcal A}^* }[/math]. A closed (in this topology) subset [math]\displaystyle{ F }[/math] of [math]\displaystyle{ \Delta {\mathcal A} }[/math] is called a boundary of [math]\displaystyle{ {\mathcal A} }[/math] if [math]\displaystyle{ \max_{f \in \Delta {\mathcal A}} |f(x)|=\max_{f \in F} |f(x)| }[/math] for all [math]\displaystyle{ x \in \mathcal A }[/math]. The set [math]\displaystyle{ S = \bigcap\{F:F \text{ is a boundary of } {\mathcal A}\} }[/math] is called the Shilov boundary. It has been proved by Shilov that [math]\displaystyle{ S }[/math] is a boundary of [math]\displaystyle{ {\mathcal A} }[/math].

Thus one may also say that Shilov boundary is the unique set [math]\displaystyle{ S \subset \Delta \mathcal A }[/math] which satisfies

1. [math]\displaystyle{ S }[/math] is a boundary of [math]\displaystyle{ \mathcal A }[/math], and
2. whenever [math]\displaystyle{ F }[/math] is a boundary of [math]\displaystyle{ \mathcal A }[/math], then [math]\displaystyle{ S \subset F }[/math].

Examples

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

References

• Hazewinkel, Michiel, ed. (2001), "Bergman-Shilov boundary", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=B/b110310 

See also

• James boundary
• Furstenberg boundary