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Gelfand-Naimark theorem (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.



"Gelfand-Naimark theorem" is owned by mhale.
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Cross-references: equivalence of categories, topology, characters, compact, contravariant functor, approximate units, *-homomorphisms, commutative, morphisms, continuous proper maps, locally compact Hausdorff spaces, category
There are 3 references to this entry.

This is version 2 of Gelfand-Naimark theorem, born on 2003-02-26, modified 2004-04-16.
Object id is 4065, canonical name is GelfandNaimarkTheorem.
Accessed 3989 times total.

Classification:
AMS MSC: 46L85 (Functional analysis :: Selfadjoint operator algebras :: Noncommutative topology)
Pending Errata and Addenda
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