Shilov boundary

Last updated November 17, 2023

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 $\mathbb {D} =\{z\in \mathbb {C$ 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 $S=\{|z|=1\}$ .

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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.

v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem