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.

1 Precise definition and existence
2 Examples
3 References
4 See also

[edit] 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 $\max_{f \in \Delta {\mathcal A}} |x(f)|=\max_{f \in S} |x(f)|$ 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

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

[edit] 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 $S = { | z | = 1}$ .