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 ${\textstyle \max _{f\in \Delta {\mathcal {A}}}|f(x)|=\max _{f\in F}|f(x)|}$ for all $x\in {\mathcal {A}}$ . The set ${\textstyle S=\bigcap \{F:F{\text{ is a boundary of }}{\mathcal {A}}\}}$ 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 $S\subset F$ .

Examples

Let $\mathbb {D} =\{z\in \mathbb {C} :|z|<1\}$ 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

"Bergman-Shilov boundary", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

Notes

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

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