Editing Gelfand representation (section)

=== 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>