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 $\mathcal A$ be a commutative Banach algebra and let $\Delta \mathcal A$ be its structure space equipped with the relative weak*-topology of the dual ${\mathcal A}^*$. A closed (in this topology) subset $F$ of $\Delta {\mathcal A}$ is called a boundary of ${\mathcal A}$ if $\max_{f \in \Delta {\mathcal A}} |x(f)|=\max_{f \in F} |x(f)|$ for all $x \in \mathcal A$. The set $S=\bigcap\{F:F \text{ is a boundary of } {\mathcal A}\}$ is called the Shilov boundary. It has been proved by Shilov^[1] that $S$ is a boundary of ${\mathcal A}$.

Thus one may also say that Shilov boundary is the unique set $S \subset \Delta \mathcal A$ which satisfies

1. $S$ is a boundary of $\mathcal A$, and
2. whenever $F$ is a boundary of $\mathcal A$, then $S \subset F$.

Examples

• Let $\mathbb D=\{z \in \mathbb C:|z|<1\}$ be the open unit disc in the complex plane and let

${\mathcal A}={\mathcal H}(\mathbb D)\cap {\mathcal C}(\bar{\mathbb D})$ be the disc algebra, i.e. the functions holomorphic in $\mathbb D$ and continuous in the closure of $\mathbb D$ with supremum norm and usual algebraic operations. Then $\Delta {\mathcal A}=\bar{\mathbb D}$ and $S=\{|z|=1\}$.

References

• Hazewinkel, Michiel, ed. (2001), "Bergman-Shilov boundary", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Notes

1. Theorem 4.15.4 in Einar Hille, Ralph S. Phillips: Functional analysis and semigroups. -- AMS, Providence 1957.

See also

• James boundary