Gelfand Representation

In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) has two related meanings:

• a way of representing commutative Banach algebras as algebras of continuous functions;
• the fact that for commutative C*-algebras, this representation is an isometric isomorphism.

In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand-Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix.