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 $\mathcal A$ 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 $\max_{f \in \Delta {\mathcal A}} |f(x)|=\max_{f \in F} |f(x)|$ 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 ShilovTheorem 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. 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 $S \subset F$ .

Examples

Let $\mathbb D=\{z \in \Complex:|z|<1\}$ be the open unit disc in the complex plane and let ${\mathcal A} = H^\infty(\mathbb D)\cap {\mathcal C}(\bar{\mathbb D})$ 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

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Shilov boundary

Precise definition and existence

Examples

References

Notes

See also