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.

Let ${\displaystyle {\mathcal {A}}}${\displaystyle {\mathcal {A}}} be a commutative Banach algebra and let ${\displaystyle \Delta {\mathcal {A}}}${\displaystyle \Delta {\mathcal {A}}} be its structure space equipped with the relative weak*-topology of the dual ${\displaystyle {\mathcal {A}}^{*}}${\displaystyle {\mathcal {A}}^{*}}. A closed (in this topology) subset ${\displaystyle F}${\displaystyle F} of ${\displaystyle \Delta {\mathcal {A}}}${\displaystyle \Delta {\mathcal {A}}} is called a boundary of ${\displaystyle {\mathcal {A}}}${\displaystyle {\mathcal {A}}} if ${\textstyle \max _{f\in \Delta {\mathcal {A}}}|f(x)|=\max _{f\in F}|f(x)|}${\textstyle \max _{f\in \Delta {\mathcal {A}}}|f(x)|=\max _{f\in F}|f(x)|} for all ${\displaystyle x\in {\mathcal {A}}}${\displaystyle x\in {\mathcal {A}}}. The set ${\textstyle S=\bigcap \{F:F{\text{ is a boundary of }}{\mathcal {A}}\}}${\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 ${\displaystyle S}${\displaystyle S} is a boundary of ${\displaystyle {\mathcal {A}}}${\displaystyle {\mathcal {A}}}.

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

1. ${\displaystyle S}${\displaystyle S} is a boundary of ${\displaystyle {\mathcal {A}}}${\displaystyle {\mathcal {A}}}, and
2. whenever ${\displaystyle F}${\displaystyle F} is a boundary of ${\displaystyle {\mathcal {A}}}${\displaystyle {\mathcal {A}}}, then ${\displaystyle S\subset F}${\displaystyle S\subset F}.

Let ${\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}}${\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}} be the open unit disc in the complex plane and let ${\displaystyle {\mathcal {A}}=H^{\infty }(\mathbb {D} )\cap {\mathcal {C}}({\bar {\mathbb {D} }})}${\displaystyle {\mathcal {A}}=H^{\infty }(\mathbb {D} )\cap {\mathcal {C}}({\bar {\mathbb {D} }})} be the disc algebra, i.e. the functions holomorphic in ${\displaystyle \mathbb {D} }${\displaystyle \mathbb {D} } and continuous in the closure of ${\displaystyle \mathbb {D} }${\displaystyle \mathbb {D} } with supremum norm and usual algebraic operations. Then ${\displaystyle \Delta {\mathcal {A}}={\bar {\mathbb {D} }}}${\displaystyle \Delta {\mathcal {A}}={\bar {\mathbb {D} }}} and ${\displaystyle S=\{|z|=1\}}${\displaystyle S=\{|z|=1\}}.

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

• James boundary
• Furstenberg boundary

• Banach algebras

• Pages that use a deprecated format of the math tags