# Gelfand-Naimark theorem

Let $\mathrm{\mathbf{H}\mathbf{a}\mathbf{u}\mathbf{s}}$ be the category^{} of locally compact Hausdorff spaces^{}
with continuous proper maps as morphisms.
And, let ${\mathbf{C}}^{*}\mathrm{\mathbf{A}\mathbf{l}\mathbf{g}}$ be the category of commutative^{} ${C}^{*}$-algebras^{}
with proper *-homomorphisms^{} (send approximate units^{} into approximate units)
as morphisms.
There is a contravariant functor^{} $C:{\mathrm{\mathbf{H}\mathbf{a}\mathbf{u}\mathbf{s}}}^{\mathrm{op}}\to {\mathbf{C}}^{*}\mathrm{\mathbf{A}\mathbf{l}\mathbf{g}}$ 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 $M:{{\mathbf{C}}^{*}\mathrm{\mathbf{A}\mathbf{l}\mathbf{g}}}^{\mathrm{op}}\to \mathrm{\mathbf{H}\mathbf{a}\mathbf{u}\mathbf{s}}$ 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^{} |
---|---|

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 |