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Gel'fand duality : ウィキペディア英語版
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
*-algebra
s, 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.
== Historical remarks ==
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 ''L''1(R) and \ell^1() whose translates span dense subspaces in the respective algebras.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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