Gelfand-Naimark theorem

Let $\mathrm{𝐇𝐚𝐮𝐬}$ be the category of locally compact Hausdorff spaceswith continuous proper maps as morphisms.And, let ${𝐂}^{*}\mathrm{𝐀𝐥𝐠}$ be the category of commutative $C^{*}$-algebraswith proper *-homomorphisms (send approximate units into approximate units)as morphisms.There is a contravariant functor $C:{\mathrm{𝐇𝐚𝐮𝐬}}^{\mathrm{op}}\to {𝐂}^{*}\mathrm{𝐀𝐥𝐠}$ 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 $M:{{𝐂}^{*}\mathrm{𝐀𝐥𝐠}}^{\mathrm{op}}\to \mathrm{𝐇𝐚𝐮𝐬}$ 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.