GelfandNaimarkTheorem 2003-02-26 17:16:10 2004-04-16 10:41:36 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % making logically defined graphics %\usepackage{xypic} % my maths package \newcommand*{\Nset}{\mathbb{N}} \newcommand*{\Zset}{\mathbb{Z}} \newcommand*{\Qset}{\mathbb{Q}} \newcommand*{\Rset}{\mathbb{R}} \newcommand*{\Cset}{\mathbb{C}} \newcommand*{\Hset}{\mathbb{H}} \newcommand*{\Oset}{\mathbb{O}} \newcommand*{\Bset}{\mathbb{B}} \newcommand*{\Kset}{\mathbb{K}} \newcommand*{\Sset}{\mathbb{S}} \newcommand*{\Tset}{\mathbb{T}} \newcommand*{\GLgrp}{\mathrm{GL}} \newcommand*{\SLgrp}{\mathrm{SL}} \newcommand*{\Ogrp}{\mathrm{O}} \newcommand*{\SOgrp}{\mathrm{SO}} \newcommand*{\Ugrp}{\mathrm{U}} \newcommand*{\SUgrp}{\mathrm{SU}} \newcommand*{\e}{\mathop{\mathrm{e}}\nolimits} \newcommand*{\im}{\mathord{\mathrm{i}}} \newcommand*{\identity}{\mathord{\mathrm{1\!\!\!\:I}}} \newcommand*{\tr}{\mathop{\mathrm{tr}}} \newcommand*{\Tr}{\mathop{\mathrm{Tr}}} \renewcommand*{\d}{\mathrm{d}} \newcommand*{\deriv}[2]{\frac{\d #1}{\d #2}} \newcommand*{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand*{\fderiv}[2]{\frac{\delta #1}{\delta #2}} % my noncommutative geometry package \newcommand*{\algebra}[1][A]{\mathord{\mathcal{#1}}} \newcommand*{\hilbert}[1][H]{\mathord{\mathcal{#1}}} \newcommand*{\hilbmod}[1][E]{\mathord{\mathcal{#1}}} \newcommand*{\Matrix}[2]{\mathord{\mathrm{M}_{#1}(#2)}} \newcommand*{\dixmier}{\mathop{\mathrm{Tr}_\omega}} \newcommand*{\Res}{\mathop{\mathrm{Res}}} \newcommand*{\Wres}{\mathop{\mathrm{Wres}}} \newcommand*{\Aut}{\mathop{\mathrm{Aut}}\nolimits} \newcommand*{\Inn}{\mathop{\mathrm{Inn}}\nolimits} \newcommand*{\Out}{\mathop{\mathrm{Out}}\nolimits} \newcommand*{\Diff}{\mathop{\mathrm{Diff}}\nolimits} \newcommand*{\Ker}{\mathop{\mathrm{Ker}}\nolimits} \newcommand*{\Coker}{\mathop{\mathrm{Coker}}\nolimits} \newcommand*{\Img}{\mathop{\mathrm{Im}}\nolimits} \newcommand*{\End}{\mathop{\mathrm{End}}\nolimits} \newcommand*{\spin}{\mathop{\mathrm{spin}}\nolimits} \newcommand*{\Ind}{\mathop{\mathrm{Ind}}\nolimits} \newcommand*{\KK}{\mathit{KK}} \newcommand*{\HH}{\mathit{HH}} \newcommand*{\HC}{\mathit{HC}} \newcommand*{\ch}{\mathop{\mathrm{ch}}\nolimits} % my category theory package \newcommand*{\mathcat}[1]{\mathord{\mathbf{#1}}} \newcommand*{\id}{\mathrm{id}} \newcommand*{\op}{\mathrm{op}} \newcommand*{\boxprod}{\mathbin{\square}} % my environments \newtheoremstyle{inlinedefn}{}{0pt}{}{}{\bfseries}{.}{0.5em}{} \theoremstyle{inlinedefn} \newtheorem{definition}{Definition} \newtheoremstyle{break}{\baselineskip}{\baselineskip}{\itshape}{}{\bfseries}{}{\newline}{} \theoremstyle{break} \newtheorem{example}{Example} % misc commands \newcommand*{\defn}[1]{\textbf{#1}} Let $\mathcat{Haus}$ be the category of locally compact Hausdorff spaces with continuous proper maps as morphisms. And, let $\mathcat{C^*Alg}$ be the category of commutative $C^*$-algebras with proper *-homomorphisms (send approximate units into approximate units) as morphisms. There is a contravariant functor $C\colon \mathcat{Haus}^\op \to \mathcat{C^*Alg}$ 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 $M\colon \mathcat{C^*Alg}^\op \to \mathcat{Haus}$ 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.