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

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

1. $S$ is a boundary of ${𝒜}$, and
2. whenever $F$ is a boundary of ${𝒜}$, 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 ${𝒜}=H^{\mathrm{\infty }}(\mathbb{𝔻})\cap {𝒞}(\overline{\mathbb{𝔻}})$ 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 $\mathrm{\Delta }{𝒜}=\overline{\mathbb{𝔻}}$ and $S=\{|z|=1\}$.

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