Spectrum of a commutative Banach algebra

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

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 ${\displaystyle \ell ^{1}({\mathbf {Z} })}$ 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:

Note that A 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.

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