functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

, a uniform algebra ''A'' on a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Hausdorff

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

''X'' is a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

(with respect to the

uniform norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the ...

)

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...

of the

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...

''C(X)'' (the

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

complex-valued In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

functions on ''X'') with the following properties: :the constant functions are contained in ''A'' : for every ''x'', ''y''

\in

''X'' there is ''f''

\in

''A'' with ''f''(''x'')

\ne

''f''(''y''). This is called separating the points of ''X''. As a closed subalgebra of the

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...

''C(X)'' a uniform algebra is itself a unital commutative Banach algebra (when equipped with the uniform norm). Hence, it is, (by definition) a Banach function algebra. A uniform algebra ''A'' on ''X'' is said to be natural if the

maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...

s of ''A'' are precisely the ideals

M_x

of functions vanishing at a point ''x'' in ''X''.

Abstract characterization

If ''A'' is a unital

such that

, , a^2, ,  = , , a, , ^2

for all ''a'' in ''A'', then there is a

Hausdorff ''X'' such that ''A'' is isomorphic as a Banach algebra to a uniform algebra on ''X''. This result follows from the

spectral radius formula In mathematics, the spectral radius of a matrix (mathematics), square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the e ...

and the

Gelfand representation In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-alge ...

Notes

References

* * Functional analysis Banach algebras {{mathanalysis-stub