Editing Gelfand representation (section)

== Gelfand representation of a commutative Banach algebra ==
Let <math> A </math> be a commutative [[Banach algebra]], defined over the field <math> \mathbb{C} </math> of complex numbers. A non-zero [[algebra homomorphism]] (a multiplicative linear functional) <math> \Phi \colon A \to \mathbb{C} </math> is called a ''character'' of <math> A </math>; the set of all characters of <math> A </math> is denoted by <math> \Phi_A </math>.

It can be shown that every character on <math> A </math> is automatically continuous, and hence <math> \Phi_A </math> is a subset of the space <math> A^* </math> of continuous linear functionals on <math> A </math>; moreover, when equipped with the relative [[Weak topology#The weak-* topology|weak-* topology]], <math> \Phi_A </math> turns out to be locally compact and Hausdorff. (This follows from the [[Banach–Alaoglu theorem]].) The space <math> \Phi_A </math> is compact (in the topology just defined) if and only if the algebra <math> A </math> has an identity element.<ref>{{citation|author=Charles Rickart|title=General theory of Banach algebras|year=1974|publisher=van Nostrand|page=114}}</ref>

Given <math> a \in A </math>, one defines the function <math>\widehat{a}:\Phi_A\to{\mathbb C}</math> by <math>\widehat{a}(\phi)=\phi(a)</math>. The definition of <math> \Phi_A </math>and the topology on it ensure that <math>\widehat{a}</math> is continuous and [[vanish at infinity|vanishes at infinity]],<ref name=":0" /> and that the map <math>a\mapsto \widehat{a}</math> defines a norm-decreasing, unit-preserving algebra homomorphism from <math> A </math> to <math> C_0(\Phi_A)</math>. This homomorphism is the ''Gelfand representation of <math> A </math>'', and <math>\widehat{a}</math> is the ''Gelfand transform'' of the element <math>a</math>. In general, the representation is neither injective nor surjective.

In the case where <math> A </math> has an identity element, there is a bijection between <math> \Phi_A </math> and the set of maximal ideals in <math> A </math> (this relies on the [[Gelfand–Mazur theorem]]). As a consequence, the kernel of the Gelfand representation <math> A \to C_0 (\Phi_A) </math> may be identified with the [[Jacobson radical]] of <math> A </math>. Thus the Gelfand representation is injective if and only if <math> A </math> is [[Semiprimitive ring|(Jacobson) semisimple]].

=== Examples ===
The Banach space <math> A=L^1(\mathbb{R})</math> is a Banach algebra under the convolution, the group algebra of <math> \mathbb{R} </math>.  Then <math> \Phi_A </math> is homeomorphic to <math> \mathbb{R} </math> and the Gelfand transform of <math> f \in L^1(\mathbb{R}) </math> is the [[Fourier transform]] <math>\tilde{f}</math>.  Similarly, with <math> A=L^1(\mathbb{R}_+)</math>, the group algebra of the multiplicative reals, the Gelfand transform is the [[Mellin transform]].

For <math>A=\ell^\infty</math>, the representation space is the [[Stone–Čech compactification]] <math>\beta\mathbb N</math>.  More generally, if <math>X</math> is a completely regular Hausdorff space, then the representation space of the Banach algebra of bounded continuous functions is the Stone–Čech compactification of <math>X</math>.<ref name=":0">Eberhard Kainuth (2009), ''A Course in Commutative Banach Algebras'', Springer</ref>