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Precise definition and existence

Let A {\displaystyle {\mathcal {A}}} be a commutative Banach algebra and let Δ A {\displaystyle \Delta {\mathcal {A}}} be its structure space equipped with the relative weak*-topology of the dual A ∗ {\displaystyle {\mathcal {A}}^{*}} . A closed (in this topology) subset F {\displaystyle F} of Δ A {\displaystyle \Delta {\mathcal {A}}} is called a boundary of A {\displaystyle {\mathcal {A}}} if max f ∈ Δ A | f ( x ) | = max f ∈ F | f ( x ) | {\textstyle \max _{f\in \Delta {\mathcal {A}}}|f(x)|=\max _{f\in F}|f(x)|} for all x ∈ A {\displaystyle x\in {\mathcal {A}}} . The set S = ⋂ { F : F  is a boundary of  A } {\textstyle S=\bigcap \{F:F{\text{ is a boundary of }}{\mathcal {A}}\}} is called the Shilov boundary. It has been proved by Shilov¹ that S {\displaystyle S} is a boundary of A {\displaystyle {\mathcal {A}}} .

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

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

Examples

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

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

Notes

See also

• James boundary
• Furstenberg boundary

References

1. Theorem 4.15.4 in Einar Hille, Ralph S. Phillips: Functional analysis and semigroups. -- AMS, Providence 1957. /wiki/Einar_Hille ↩