Gelfand-Naimark theorem

In functional analysis,Gel'fand representation allowscomplete characterisationcommutative C-star-algebras. Thisone terminal point indevelopmentspectral theorynormal operators.

For any compact Hausdorff topological space X,space C(X)continuous complex-valued functions on X becomescommutative C*-algebra fornatural ring structure anduniform norm on functions. Conversely given such an algebra A, one can constructspace Yall maximal ideals mA, withsuitable topology. For any such m itshown that A/mnaturally identified withcomplex numbers C. Therefore anyin A gives rise tocomplex-valued function on Y.

The content ofGel'fand representation theoremthatthis way A becomes isomorphicC(Y), Y indeed being compactHausdorff. We can call YspectrumA. Further we havecontravariant functor: morphismsC*-algebras give risecontinuous mapsspectrum spaces, inother direction.