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