Gelfand-Naimark theorem

Let 𝐇𝐚𝐮𝐬 be the categoryMathworldPlanetmath of locally compact Hausdorff spacesPlanetmathPlanetmath with continuous proper maps as morphisms. And, let 𝐂*𝐀𝐥𝐠 be the category of commutativePlanetmathPlanetmathPlanetmath C*-algebrasPlanetmathPlanetmath with proper *-homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (send approximate unitsMathworldPlanetmath into approximate units) as morphisms. There is a contravariant functorMathworldPlanetmath C:𝐇𝐚𝐮𝐬op𝐂*𝐀𝐥𝐠 which sends each locally compact Hausdorff space X to the commutative C*-algebra C0(X) (C(X) if X is compactPlanetmathPlanetmath). Conversely, there is a contravariant functor M:𝐂*𝐀𝐥𝐠op𝐇𝐚𝐮𝐬 which sends each commutative C*-algebra A to the space of charactersMathworldPlanetmathPlanetmath on A (with the Gelfand topologyMathworldPlanetmath).

The functors C and M are an equivalence of categories.

Title Gelfand-Naimark theoremMathworldPlanetmathPlanetmath
Canonical name GelfandNaimarkTheorem
Date of creation 2013-03-22 13:29:28
Last modified on 2013-03-22 13:29:28
Owner mhale (572)
Last modified by mhale (572)
Numerical id 5
Author mhale (572)
Entry type Theorem
Classification msc 46L85