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*-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.