\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Gelfand duality}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - contents]]

\hypertarget{noncommutative_geometry}{}\paragraph*{{Noncommutative geometry}}\label{noncommutative_geometry}

[[!include noncommutative geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{in_constructive_mathematics}{In constructive mathematics}\dotfill \pageref*{in_constructive_mathematics} \linebreak
\noindent\hyperlink{by_horizontal_categorification}{By horizontal categorification}\dotfill \pageref*{by_horizontal_categorification} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Gelfand duality} is a [[Isbell duality|duality between spaces and their algebras of functions]] for the case of [[compact space|compact]] [[topological space]]s and [[commutative C-star-algebra|commutative]] [[C-star algebras]]:

every (nonunital) $C^\ast$-algebra $A$ is equivalent to the $C^\ast$-algebra of [[continuous functions]] on the [[topological space]] called its \emph{[[Gelfand spectrum]]} $sp(A)$.

This theorem is the basis for regarding non-commutative $C^\ast$-algebras as formal duals to spaces in [[noncommutative geometry]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

The statement of Gelfand duality involves the following [[categories]] and [[functors]].

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{itemize}%
\item $C^\ast Alg$ for the category of [[C-star algebras]];


\item $C^\ast Alg_{nu}$ for the category of non-unital $C^\ast$-algebras;


\item $C^\ast Alg_{com} \subset C^\ast Alg$ for the [[full subcategory]] of commutative $C^\ast$-algebras;


\item $C^\ast Alg_{com,nu} \subset C^\ast Alg_{nu}$ for the full subcategory of commutative non-unital $C^\ast$-algebras.



\end{itemize}
And

\begin{itemize}%
\item [[Top]]${}_{Haus}$ for the category of [[Hausdorff topological spaces]]


\item $Top_{cpt}$ for the [[full subcategory]] of [[Top]]${}_{Haus}$ of the [[compact topological spaces]];


\item $*/Top_{cpt}$ for the category of [[pointed topological space|pointed topological]] [[compact Hausdorff spaces]], i.e. the [[pointed objects]] in $Top_{cpt}$;


\item $Top_{lcpt}$ for the category of Hausdorff and [[locally compact topological spaces]] with morphisms being the [[proper maps]] of topological spaces.



\end{itemize}
\end{defn}
The duality itself is exhibited by the following [[functors]]:

\begin{defn}
\label{FunctorsOfFunctionAlgebras}\hypertarget{FunctorsOfFunctionAlgebras}{}
Write

\begin{displaymath}
C \;\colon\; Top_{cpt} \to C^\ast Alg_{com}^{op}
\end{displaymath}
for the functor which sends a [[compact topological space]] $X$ to the algebra of [[continuous function]]s $C(X) = \{f : X \to \mathbb{C} | f \; continuous\}$, equipped with the structure of a $C^\ast$-algebra in the evident way (\ldots{}).

Write

\begin{displaymath}
C_0 \;\colon\; */Top_{cpt} \to C^\ast Alg_{com,nu}
\end{displaymath}
for the functor that sends a [[pointed topological space|pointed topological]] [[compact Hausdorff space]] $(X,x_0)$ to the algebra of [[continuous function]]s $f : X \to \mathbb{C}$ for which $f(x_0) = 0$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
sp : C^\ast Alg_{com}^{op} \to Top_{cpt}
\end{displaymath}
for the [[Gelfand spectrum]] functor: it sends a commutative $C^\ast$-algebra $A$ to the set of \emph{[[character]]s} -- non-vanishing $C^\ast$-algebra [[homomorphisms]] $x : A \to \mathbb{C}$ -- equipped with the [[spectral topology]].

Similarly write

\begin{displaymath}
sp : C^\ast Alg_{com,nu}^{op} \to Top_{lcpt}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{theorem}
\label{GelfandDualityTheorem}\hypertarget{GelfandDualityTheorem}{}
\textbf{(Gelfand duality theorem)}

The pairs of functors

\begin{displaymath}
C^\ast Alg_{com}^{op} \stackrel{\overset{C}{\leftarrow}}{\underset{sp}{\to}} Top_{cpt}
\end{displaymath}
is an [[equivalence of categories]].

\end{theorem}
Here $C^\ast Alg^{op}_{\cdots}$ denotes the [[opposite category]] of $C^\ast Alg_{\cdots}$.

\begin{cor}
\label{}\hypertarget{}{}
On non-unital $C^\ast$-algebras the above induces an [[equivalence of categories]]

\begin{displaymath}
C^\ast Alg_{com,nu}^{op} \stackrel{\overset{C_0}{\leftarrow}}{\underset{sp}{\to}} */Top_{cpt}
  \,.
\end{displaymath}
\end{cor}
\begin{proof}
The operation of [[unitalization]] $(-)^+$ constitutes an [[equivalence of categories]]

\begin{displaymath}
C^\ast Alg_{nu}
    \stackrel{\overset{ker}{\leftarrow}}{\underset{(-)^+}{\to}}
  C^\ast Alg / \mathbb{C}
\end{displaymath}
between non-unital $C^\ast$-algebras and the [[over-category]] of $C^\ast$-algebras over the [[complex number]]s $\mathbb{C}$.

Composed with the equivalence of theorem \ref{GelfandDualityTheorem} this yields

\begin{displaymath}
C^\ast Alg_{com,nu}^{op}
    \underoverset{\simeq}{(-)^+}{\to}
  (C^\ast Alg_{com}/\mathbb{C})^{op}
    \underoverset{\simeq}{C}{\to}
  * / Top_{cpt}
  \,.
\end{displaymath}
The weak inverse of this is the composite functor

\begin{displaymath}
C_0 : */Top_{cpt} 
    \underoverset{\simeq}{sp}{\to}
  (C^\ast Alg_{com}/\mathbb{C})^{op}
    \underoverset{\simeq}{ker}{\to}
  C^\ast Alg_{com,nu}^{op}
\end{displaymath}
which sends $(* \stackrel{x_0}{\to} X)$ to $ker(C(X) \stackrel{ev_{x_0}}{\to} \mathbb{C})$, hence to $\{f \in C(X) | f(x_0) = 0\}$. This is indeed $C_0$ from def. \ref{FunctorsOfFunctionAlgebras}.

\end{proof}
\begin{remark}
\label{ForLocallyCompactTopologicalSpaces}\hypertarget{ForLocallyCompactTopologicalSpaces}{}
Since [[locally compact Hausdorff spaces are equivalently open subspaces of compact Hausdorff spaces]], via the construction that sends a locally compact Hausdorff space $X$ to its [[one-point compactification]], and since a [[continuous function]] on the compact Hausdorff space $X^\ast$ which vanishes at the extra point is equivalently a continuous function on $X$ which [[vanishing at infinity|vanishes at infinity]], the above induces an equivalence between locally compact Hausdorff spaces and $C^\ast$-algebras of functions that vanish at infinity.

With due care on defining the right morphisms, the duality generalizes also to [[locally compact topological spaces]]. See for instance (\hyperlink{Brandenburg07}{Brandenburg 07}).

The duality also works with [[real numbers]] instead of [[complex numbers]] (\hyperlink{Johnstone82}{Johnstone 82, chapter IV})

For an overview of other generalizations see also \href{http://mathoverflow.net/a/82960/381}{this MO discussion}.

\end{remark}
\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\hypertarget{in_constructive_mathematics}{}\subsubsection*{{In constructive mathematics}}\label{in_constructive_mathematics}

Gelfand duality makes sense in [[constructive mathematics]] hence [[internalization|internal]] to any [[topos]]: see [[constructive Gelfand duality theorem]].

\hypertarget{by_horizontal_categorification}{}\subsubsection*{{By horizontal categorification}}\label{by_horizontal_categorification}

Gelfand duality can be extended by [[horizontal categorification]] to define the notion of [[spaceoid|spaceoids]] as formal duals of commutative $C^*$-[[C-star-category|categories]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Isbell duality]]

\begin{itemize}%
\item [[Gelfand-Kolmogorov theorem]]


\item \textbf{Gelfand duality}, [[constructive Gelfand duality theorem|constructive Gelfand duality]]

\begin{itemize}%
\item [[C-star algebra]], [[topological space]]


\item [[Gelfand-Naimark theorem]]


\item [[Gelfand spectrum]]


\item [[Serre-Swan theorem]]


\item [[noncommutative topology]]



\end{itemize}


\end{itemize}

\item [[Stone duality]]



\end{itemize}
The analogous statement in [[differential geometry]]:

\begin{itemize}%
\item [[embedding of smooth manifolds into formal duals of R-algebras]]

\end{itemize}
[[!include Isbell duality - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Formulation of Gelfand duality in terms of [[category theory]] ([[adjoint functors]], [[monads]] (``triples'') and [[adjoint equivalences]]) originates with

\begin{itemize}%
\item Joan W. Negrepontis, \emph{Duality in analysis from the point of view of triples}, Journal of Algebra, 19 (2): 228–253, (1971) (, ISSN 0021-8693, MR 0280571)

\end{itemize}
Quick exposition is in

\begin{itemize}%
\item Ivo Dell'Ambrogio, \emph{Categories of $C^\ast$-algebras} (\href{http://www.math.uni-bielefeld.de/~ambrogio/exercise_C_algebras.pdf}{pdf})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Peter Johnstone]], section IV.4 of \emph{[[Stone Spaces]]}, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press 1982. xxi+370 pp. MR85f:54002, reprinted 1986.


\item [[Nicolaas Landsman|N. P. Landsman]], \emph{Mathematical topics between classical and quantum mechanics}, Springer Monographs in Mathematics 1998. xx+529 pp. \href{http://www.ams.org/mathscinet-getitem?mr=1662141}{MR2000g:81081} \href{http://dx.doi.org/10.1007/978-1-4612-1680-3}{doi}


\item Gerald B. Folland, \emph{A course in abstract harmonic analysis}, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. \href{http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC}{gBooks}



\end{itemize}
Careful discussion of the duality for the more general case of [[locally compact topological spaces]] includes

\begin{itemize}%
\item [[Martin Brandenburg]], \emph{Gelfand-Dualit\"a{}t ohne 1}, 2007 (\href{http://www.matheplanet.com/matheplanet/nuke/html/article.php?sid=1111}{web})

\end{itemize}
 Discussion of Gelfand duality as a [[fixed point equivalence of an adjunction]] between includes

\begin{itemize}%
\item [[Hans-E. Porst]], [[Walter Tholen]], section 4-c of \emph{Concrete Dualities} in H. Herrlich, [[Hans-E. Porst]] (eds.) \emph{Category Theory at Work}, Heldermann Verlag 1991 (\href{http://www.heldermann.de/R&E/RAE18/ctw07.pdf}{pdf})

\end{itemize}
following

\begin{itemize}%
\item [[Eduardo Dubuc]], [[Horacio Porta]], \emph{Convenient categories of topological algebras}, Bull. Amer. Math. Soc., Volume 77, Number 6 (1971), 975-979 \href{https://projecteuclid.org/euclid.bams/1183533170}{euclid}

\end{itemize}
Some other generalized contexts for Gelfand duality:

\begin{itemize}%
\item [[Hans Porst]], Manfred B. Wischnewsky, \emph{Every topological category is convenient for Gelfand duality}, Manuscripta mathematica \textbf{25}:2, (1978) pp 169-204


\item [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], Sander Wolters, \emph{The Gelfand spectrum of a noncommutative $C^\ast$-algebra}, J. Aust. Math. Soc. \textbf{90} (2011), 39--52 \href{http://dx.doi.org/10.1017/S1446788711001157}{doi} \href{http://www.math.ru.nl/~landsman/LandsmanCarey60.pdf}{pdf}


\item [[Christopher J. Mulvey]], \emph{A generalisation of Gelfand duality}, J. Algebra 56, n. 2, (1979) 499--505 


\item [[Arthur Parzygnat]], \emph{Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem} (\href{https://arxiv.org/abs/1708.00091}{arXiv:1708.00091})



\end{itemize}
category: analysis, geometry, noncommutative geometry

[[!redirects Gelfand duality]] [[!redirects Gel'fand duality]]

[[!redirects Gelfand-Naimark duality]]



\end{document}