# Gelfand-Naimark theorem

Let $\mathord{\mathbf{Haus}}$ be the category of locally compact Hausdorff spaces with continuous proper maps as morphisms. And, let $\mathord{\mathbf{C^{*}Alg}}$ be the category of commutative $C^{*}$-algebras with proper *-homomorphisms (send approximate units into approximate units) as morphisms. There is a contravariant functor $C\colon\mathord{\mathbf{Haus}}^{\mathrm{op}}\to\mathord{\mathbf{C^{*}Alg}}$ which sends each locally compact Hausdorff space $X$ to the commutative $C^{*}$-algebra $C_{0}(X)$ ($C(X)$ if $X$ is compact). Conversely, there is a contravariant functor $M\colon\mathord{\mathbf{C^{*}Alg}}^{\mathrm{op}}\to\mathord{\mathbf{Haus}}$ which sends each commutative $C^{*}$-algebra $A$ to the space of characters on $A$ (with the Gelfand topology).

The functors $C$ and $M$ are an equivalence of categories.

Title Gelfand-Naimark theorem GelfandNaimarkTheorem 2013-03-22 13:29:28 2013-03-22 13:29:28 mhale (572) mhale (572) 5 mhale (572) Theorem msc 46L85