Banach function algebra

In functional analysis, a Banach function algebra on a compact Hausdorff space X is unital subalgebra, A, of the commutative C*-algebra C(X) of all continuous, complex-valued functions from X, together with a norm on A that makes it a Banach algebra. A function algebra is said to vanish at a point p if f(p) = 0 for all f ∈ A {\displaystyle f\in A} . A function algebra separates points if for each distinct pair of points p , q ∈ X {\displaystyle p,q\in X} , there is a function f ∈ A {\displaystyle f\in A} such that f ( p ) ≠ f ( q ) {\displaystyle f(p)\neq f(q)} . For every x ∈ X {\displaystyle x\in X} define ε x ( f ) = f ( x ) , {\displaystyle \varepsilon _{x}(f)=f(x),} for f ∈ A {\displaystyle f\in A} . Then ε x {\displaystyle \varepsilon _{x}} is a homomorphism (character) on A {\displaystyle A} , non-zero if A {\displaystyle A} does not vanish at x {\displaystyle x} . Theorem: A Banach function algebra is semisimple (that is its Jacobson radical is equal to zero) and each commutative unital, semisimple Banach algebra is isomorphic (via the Gelfand transform) to a Banach function algebra on its character space (the space of algebra homomorphisms from A into the complex numbers given the relative weak* topology). If the norm on A {\displaystyle A} is the uniform norm (or sup-norm) on X {\displaystyle X} , then A {\displaystyle A} is called a uniform algebra. Uniform algebras are an important special case of Banach function algebras.

In functional analysis, a Banach function algebra on a compact Hausdorff space X is unital subalgebra, A, of the commutative C*-algebra C(X) of all continuous, complex-valued functions from X, together with a norm on A that makes it a Banach algebra.

A function algebra is said to vanish at a point p if f(p) = 0 for all $f\in A$ . A function algebra separates points if for each distinct pair of points $p,q\in X$ , there is a function $f\in A$ such that $f(p)\neq f(q)$ .

For every $x\in X$ define $\varepsilon _{x}(f)=f(x),$ for $f\in A$ . Then $\varepsilon _{x}$ is a homomorphism (character) on $A$ , non-zero if $A$ does not vanish at $x$ .

Theorem: A Banach function algebra is semisimple (that is its Jacobson radical is equal to zero) and each commutative unital, semisimple Banach algebra is isomorphic (via the Gelfand transform) to a Banach function algebra on its character space (the space of algebra homomorphisms from A into the complex numbers given the relative weak* topology).

If the norm on $A$ is the uniform norm (or sup-norm) on $X$ , then $A$ is called a uniform algebra. Uniform algebras are an important special case of Banach function algebras.

References

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Andrew Browder (1969) Introduction to Function Algebras, W. A. Benjamin
H.G. Dales (2000) Banach Algebras and Automatic Continuity, London Mathematical Society Monographs 24, Clarendon Press ISBN 0-19-850013-0
Graham Allan & H. Garth Dales (2011) Introduction to Banach Spaces and Algebras, Oxford University Press ISBN 978-0-19-920654-4

Functional analysis (topics – glossary)

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

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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 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

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