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Gelfand-Naimark theorem

In functional analysis, the Gel'fand representation allows a complete characterisation of commutative C-star-algebras. This is one terminal point in the development of spectral theory for normal operators.

For any compact Hausdorff topological space X, the space C(X) of continuous complex-valued functions on X becomes a commutative C*-algebra for the natural ring structure and the uniform norm on functions. Conversely given such an algebra A, one can construct the space Y of all maximal ideals m of A, with a suitable topology. For any such m it is shown that A/m is naturally identified with the complex numbers C. Therefore any a in A gives rise to a complex-valued function on Y.

The content of the Gel'fand representation theorem is that in this way A becomes isomorphic with C(Y), Y indeed being compact and Hausdorff. We can call Y the spectrum of A. Further we have a contravariant functor: morphisms of C*-algebras give rise to continuous maps of spectrum spaces, in the other direction.