Editing Fourier transform (section)

=== Gelfand transform ===
{{Main|Gelfand representation}}
The Fourier transform is also a special case of [[Gelfand transform]]. In this particular context, it is closely related to the Pontryagin duality map defined above.

Given an abelian [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] {{mvar|G}}, as before we consider space {{math|''L''<sup>1</sup>(''G'')}}, defined using a Haar measure. With convolution as multiplication, {{math|''L''<sup>1</sup>(''G'')}} is an abelian [[Banach algebra]]. It also has an [[Involution (mathematics)|involution]] * given by
<math display="block">f^*(g) = \overline{f\left(g^{-1}\right)}.</math>

Taking the completion with respect to the largest possibly {{math|''C''*}}-norm gives its enveloping {{math|''C''*}}-algebra, called the group {{math|''C''*}}-algebra {{math|''C''*(''G'')}} of {{mvar|G}}. (Any {{math|''C''*}}-norm on {{math|''L''<sup>1</sup>(''G'')}} is bounded by the {{math|''L''<sup>1</sup>}} norm, therefore their supremum exists.)

Given any abelian {{math|''C''*}}-algebra {{mvar|A}}, the Gelfand transform gives an isomorphism between {{mvar|A}} and {{math|''C''<sub>0</sub>(''A''^)}}, where {{math|''A''^}} is the multiplicative linear functionals, i.e. one-dimensional representations, on {{mvar|A}} with the weak-* topology. The map is simply given by
<math display="block">a \mapsto \bigl( \varphi \mapsto \varphi(a) \bigr)</math>
It turns out that the multiplicative linear functionals of {{math|''C''*(''G'')}}, after suitable identification, are exactly the characters of {{mvar|G}}, and the Gelfand transform, when restricted to the dense subset {{math|''L''<sup>1</sup>(''G'')}} is the Fourier–Pontryagin transform.