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Uniform algebra

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Mathematical concept

Infunctional analysis, auniform algebraA on acompact Hausdorff topological spaceX is aclosed (with respect to theuniform norm)subalgebra of theC*-algebraC(X) (thecontinuous complex-valued functions onX) with the following properties:^[1]

the constant functions are contained inA

for everyx,y

\in

X there isf

\in

A withf(x)

\neq

f(y). This is called separating the points ofX.

As a closed subalgebra of thecommutative Banach algebraC(X) a uniform algebra is itself a unital commutative Banach algebra (when equipped with the uniform norm). Hence, it is, (by definition) aBanach function algebra.

A uniform algebraA onX is said to benatural if themaximal ideals ofA are precisely the ideals $M_{x}$ of functions vanishing at a pointx inX.

Abstract characterization

[edit]

IfA is aunital commutative Banach algebra such that $||a^{2}||=||a||^{2}$ for alla inA, then there is acompact HausdorffX such thatA is isomorphic as a Banach algebra to a uniform algebra onX. This result follows from thespectral radius formula and theGelfand representation.

Notes

[edit]

^(Gamelin 2005, p. 25)

References

[edit]

Gamelin, Theodore W. (2005).Uniform Algebras. American Mathematical Soc.ISBN 978-0-8218-4049-8.
Gorin, E.A. (2001) [1994],"Uniform algebra",Encyclopedia of Mathematics,EMS Press

Functional analysis (topics –glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic /Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

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Advanced topics

Category

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 anApproximate 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

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