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. One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener (see the citation below), characterizing the elements of the group algebras L1(R) and whose translates span dense subspaces in the respective algebras. For any locally compact Hausdorff topological space X, the space C0(X) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C*-algebra: The structure of algebra over the complex numbers is obtained by considering the pointwise operations of addition and multiplication. The involution is pointwise complex conjugation. The norm is the uniform norm on functions. The importance of X being locally compact and Hausdorff is that this turns X into a completely regular space. In such a space every closed subset of X is the common zero set of a family of continuous complex-valued functions on X, allowing one to recover the topology of X from C0(X). Note that C0(X) is unital if and only if X is compact, in which case C0(X) is equal to C(X), the algebra of all continuous complex-valued functions on X. Let be a commutative Banach algebra, defined over the field of complex numbers. A non-zero algebra homomorphism (a multiplicative linear functional) is called a character of ; the set of all characters of is denoted by .