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\section*{C-star-algebra}

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

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

[[!include higher algebra - contents]]

\hypertarget{operator_algebra}{}\paragraph*{{Operator algebra}}\label{operator_algebra}

[[!include AQFT and operator algebra contents]]

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

[[!include noncommutative geometry - contents]]

\hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory}

[[!include index theory - contents]]

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

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{abstract_algebras}{Abstract $C^\ast$-algebras}\dotfill \pageref*{abstract_algebras} \linebreak
\noindent\hyperlink{concrete_algebras_and_representations}{Concrete $C^\ast$-algebras and $C^\ast$-representations}\dotfill \pageref*{concrete_algebras_and_representations} \linebreak
\noindent\hyperlink{DaggerFormulation}{In $\dagger$-compact categories}\dotfill \pageref*{DaggerFormulation} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{gelfandnaimark_theorem}{Gelfand-Naimark theorem}\dotfill \pageref*{gelfandnaimark_theorem} \linebreak
\noindent\hyperlink{gelfandnaimarksegal_construction}{Gelfand-Naimark-Segal construction}\dotfill \pageref*{gelfandnaimarksegal_construction} \linebreak
\noindent\hyperlink{gelfand_duality}{Gelfand duality}\dotfill \pageref*{gelfand_duality} \linebreak
\noindent\hyperlink{General}{General}\dotfill \pageref*{General} \linebreak
\noindent\hyperlink{construction_as_groupoid_convolution_algebras}{Construction as groupoid convolution algebras}\dotfill \pageref*{construction_as_groupoid_convolution_algebras} \linebreak
\noindent\hyperlink{homotopy_theory}{Homotopy theory}\dotfill \pageref*{homotopy_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{abstract_algebras}{}\subsubsection*{{Abstract $C^\ast$-algebras}}\label{abstract_algebras}

\begin{defn}
\label{}\hypertarget{}{}
A $C^*$-algebra is a [[Banach algebra]] $(A, {\|-\|})$ over a [[topological field]] $K$ (often the field $K \coloneqq \mathbb{C}$ of [[complex numbers]]) equipped with an [[involution]] $(-)^\ast$ compatible with [[complex conjugation]] if appropriate (that is: a Banach [[star-algebra]]) that satisfies the \textbf{$C^*$-identity}

\begin{displaymath}
{\|{A^* A}\|} = {\|{A^*}\|} \, {\|{A}\|}
\end{displaymath}
or equivalently the \textbf{$B^*$-identity}

\begin{displaymath}
{\|{A^* A}\|} = {\|{A}\|^2} 
  \,.
\end{displaymath}
A [[homomorphism]] of $C^\ast$-algebras is a map that preserves all this structure. For this it is sufficient for it to be a [[star-algebra]] homomorphisms.

$C^\ast$-algebras with these homomorphisms form a [[category]] [[C\emph{Alg]].}

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Often one sees the definition without the clause (which should be in the definition of Banach $*$-algebra) that the involution is an [[isometry]] (so that ${\|A^*\|} = {\|A\|}$, which is key for the equivalence of the $B^*$ and $C^*$ identities). This follows easily from the $B^*$-identity, while it follows from the $C^*$-identity after some difficulty.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
There are different concepts for the [[tensor product]] of $C^*$-algebras, see for example at \emph{[[spatial tensor product]]}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
$C^*$-algebras equipped with the [[action]] of a [[group]] by [[automorphisms]] of the algebra are called \emph{[[C-star-systems]]} .

\end{remark}
\hypertarget{concrete_algebras_and_representations}{}\subsubsection*{{Concrete $C^\ast$-algebras and $C^\ast$-representations}}\label{concrete_algebras_and_representations}

\begin{defn}
\label{}\hypertarget{}{}
Given a [[complex numbers|complex]] [[Hilbert space]] $H$, a \textbf{[[concrete structure|concrete]] $C^*$-algebra} on $H$ is a $*$-[[subalgebra]] of the algebra of [[bounded operators]] on $H$ that is [[closed subspace|closed]] in the [[norm topology]].

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{[[representation of a C-star algebra|representation]]} of a $C^*$-algebra $A$ on a [[Hilbert space]] $H$ is a $*$-homomorphism from $A$ to the algebra of [[bounded operators]] on $H$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
It is immediate that concrete $C^*$-algebras correspond precisely to [[faithful representations]] of abstract $C^*$-algebras. It is an important theorem that \emph{every} $C^*$-algebra has a faithful representation; that is, every abstract $C^*$-algebra is [[isomorphism|isomorphic]] to a concrete $C^*$-algebra.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The original definition of the term `$C^*$-algebra' was in fact the concrete notion; the `C' stood for `closed'. Furthermore, the original term for the abstract notion was `$B^*$-algebra' (where the `B' stood for `Banach'). However, we now usually interpret `$C^*$-algebra' abstractly. (Compare `$W^*$-algebra' and `[[von Neumann algebra]]'.)

\end{remark}
\hypertarget{DaggerFormulation}{}\subsubsection*{{In $\dagger$-compact categories}}\label{DaggerFormulation}

The notion of $C^*$-algebra can be abstracted to the general context of [[symmetric monoidal †-categories]], which serves to illuminate their role in \emph{[[quantum mechanics in terms of †-compact categories]]}.

For a discussion of this in the finite-dimensional case see for instance (\hyperlink{Vicary}{Vicary}).

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

$C^*$-algebras are [[monadic]] over sets. More precisely, the forgetful functor $\mathbf{C^*Alg}\to\mathbf{Set}$ that assigns to each algebra the set of points in its unit ball is monadic. See J Wick Pelletier \& J Rosicky, On the equational theory of $C^*$-algebras, Algebra Universalis 30:275-284, 1993.

See also [[operator algebras]].

\hypertarget{gelfandnaimark_theorem}{}\subsubsection*{{Gelfand-Naimark theorem}}\label{gelfandnaimark_theorem}

The \emph{[[Gelfand-Naimark theorem]]} says that every [[C\emph{-algebra]] is [[isomorphism|isomorphic]] to a $C^\ast$-algebra of [[bounded linear operators]] on a [[Hilbert space]]. In other words, every abstract $C^*$-algebra may be made into a concrete $C^*$-algebra.}

\hypertarget{gelfandnaimarksegal_construction}{}\subsubsection*{{Gelfand-Naimark-Segal construction}}\label{gelfandnaimarksegal_construction}

The [[Gelfand-Naimark-Segal construction]] ([[GNS construction]]) establishes a correspondence between cyclic $*$-[[representation]]s of $C^*$-[[C\emph{-algebra|algebras]] and certain linear functionals (usually called \emph{[[state on an operator algebra|states]]}) on those same $C^*$-algebras. The correspondence comes about from an explicit construction of the [[star-representation|}-representation]] from one of the [[linear functionals|linear functionals]] (states).

\hypertarget{gelfand_duality}{}\subsubsection*{{Gelfand duality}}\label{gelfand_duality}

[[Gelfand duality]] says that every ([[unital algebra|unital]]) \emph{[[commutative C-star-algebra|commutative]]} $C^*$-algebra over the [[complex numbers]] is that of complex-valued [[continuous functions]] from some [[compactum|compact Hausdorff topological space]]: there is an [[equivalence of categories]] $C^* CAlg \simeq$ [[Top]]${}_{cpt}$.

Accordingly one may think of the study of non-commutative $C^\ast$-algebras as \emph{[[non-commutative topology]]}.

\hypertarget{General}{}\subsubsection*{{General}}\label{General}

\begin{prop}
\label{}\hypertarget{}{}
For $A$ and $B$ two $C^\ast$-algebras and $f : A \to B$ a [[star-algebra]] [[homomorphism]] the set-theoretic [[image]] $f(A) \subset B$ is a $C^\ast$-subalgebra of $B$, hence is also the [[image]] of $f$ in $C^\ast Alg$.

\end{prop}
This is (\hyperlink{KadisonRingrose}{KadisonRingrose, theorem 4.1.9}).

\begin{cor}
\label{}\hypertarget{}{}
There is a [[functor]]

\begin{displaymath}
\mathcal{C} : C^\ast Alg \to Poset
\end{displaymath}
to the [[category]] [[Poset]] of [[posets]], which sends each $A \in C^\ast Alg$ to its [[poset of commutative subalgebras]] $\mathcal{C}(A)$ and sends each morphism $f : A \to B$ to the [[functor]] $\mathcal{C}(f) : \mathcal{C}(A) \to \mathcal{C}(B)$ which sends a [[commutative C-star-algebra|commutative]] subalgebra $C \subset A$ to $f(C) \subset B$.

\end{cor}
\hypertarget{construction_as_groupoid_convolution_algebras}{}\subsubsection*{{Construction as groupoid convolution algebras}}\label{construction_as_groupoid_convolution_algebras}

Many $C^\ast$-algebras arise as [[groupoid algebras]] of [[Lie groupoids]]. See at \emph{\href{category+algebra#ReferencesForSmoothGeometry}{groupoid algebra - References - For smooth geometry}}

\hypertarget{homotopy_theory}{}\subsubsection*{{Homotopy theory}}\label{homotopy_theory}

There is [[homotopy theory]] of $C^\ast$-algebras, being a non-commutative generalization of that of [[Top]]. (e.g. \hyperlink{Uuye}{Uuye 12}). For more see at \emph{[[homotopical structure on C\emph{-algebras]]\_.}}

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
Any algebra $M_n(A)$ of [[matrices]] with [[coefficients]] in a $C^\ast$-algebra is again a $C^\ast$-algebra. In particular $M_n(\mathbb{C})$ is a $C^\ast$-algebra for all $n \in \mathbb{N}$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $A$ a $C^\ast$-algebra and for $X$ a [[locally compact topological space|locally compact]] [[Hausdorff topological space]], the set of [[continuous functions]] $X \to A$ which [[vanish at infinity]] is again a $C^\ast$-algebra by extending all operations pointwise. (This algebra is [[unital algebra|unital]] precisely if $A$ is and if $X$ is a [[compact topological space]].)

This algebra is denoted

\begin{displaymath}
C_0(X,A) \in C^\ast Alg
  \,.
\end{displaymath}
If $A = \mathbb{C}$ then one usually just writes

\begin{displaymath}
C_0(X) \coloneqq C_0(X, \mathbb{C})
  \,.
\end{displaymath}
This are the $C^\ast$-algebras to which the [[Gelfand duality]] theorem applies.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
A [[uniformly hyperfinite algebra]] is in particular a $C^\ast$-algebra, by definition.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
A [[von Neumann algebra]] is in particular a $C^\ast$-algebra, by definition.

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

\begin{itemize}%
\item [[commutative C-star-algebra]], [[Gelfand duality]], [[noncommutative topology]]


\item [[separable C\emph{-algebra]], [[homogeneous C}-algebra]]


\item [[nuclear C\emph{-algebra]]}


\item [[unitisation of C\emph{-algebras]]}


\item [[von Neumann algebra]], [[enveloping von Neumann algebra]]


\item [[JB-algebra]], [[JLB-algebra]]


\item [[dense subalgebra]], [[F-star-algebra]]


\item [[multiplier algebra]]


\item [[graded C\emph{-algebra]]}


\item [[tensor product of C\emph{-algebras]]}


\item [[crossed product C\emph{-algebra]]}


\item [[Cuntz algebra]]


\item [[Poincaré duality C\emph{-algebra]]}


\item [[Hilbert C\emph{-module]], [[Hilbert C}-bimodule]], [[amplimorphism]]


\item [[C\emph{-category]]}


\item [[C\emph{-coalgebra]], [[Hopf C}-algebra]]


\item [[continuous field of C\emph{-algebras]]}


\item [[homotopical structure on C\emph{-algebras]]}

\begin{itemize}%
\item [[asymptotic C\emph{-homomorphism]], [[l.m.c.-C}-algebra]]


\item [[KK-theory]], [[E-theory]]



\end{itemize}

\item [[AQFT]]

\begin{itemize}%
\item [[state on an operator algebra]]

\end{itemize}


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

A standard textbook reference is chapter 4 in volume 1 of

\begin{itemize}%
\item Richard Kadison, John Ringrose, \emph{Fundamentals of the theory of operator algebras} Academic Press, (1983)

\end{itemize}
An exposition that explicitly gives [[Gelfand duality]] as an [[equivalence of categories]] and introduces all the notions of [[category theory]] necessary for this statement is in

\begin{itemize}%
\item Ivo Dell'Ambrogio, \emph{Categories of $C^\ast$-algebras} (\href{http://www.math.ethz.ch/u/ambrogio/exercise_C_-algebras.pdf}{pdf})

\end{itemize}
For [[operator algebra]]-theory see there and see

\begin{itemize}%
\item [[Stanisław Woronowicz]], \emph{Unbounded elements affiliated with $C^\ast$-algebras and non-compact quantum groups. Commun. Math. Phys. 136, 399--432 (1991)}


\item [[Stanisław Woronowicz]], K. Napi\'o{}rkowski, \emph{[[Operator theory in the C\emph{-algebra framework]]\_, Reports on Mathematical Physics Volume 31, Issue 3, June 1992, Pages 353--371 (\href{http://www.sciencedirect.com/science/article/pii/003448779290025V}{publisher}, \href{http://www.fuw.edu.pl/~slworono/PDF-y/OP.pdf}{pdf})}}



\end{itemize}
A characterizations of injections of commutative sub-$C^*$-algebras -- hence of the [[poset of commutative subalgebras]] of a $C^*$-algebra -- is in

\begin{itemize}%
\item [[Chris Heunen]], \emph{Characterizations of categories of commutative $C^*$-algebras} (\href{http://arxiv.org/abs/1106.5942}{arXiv:1106.5942})

\end{itemize}
General properties of the [[category]] of $C^\ast$-algebras are discussed in

\begin{itemize}%
\item [[Ralf Meyer]], \emph{Categorical aspects of bivariant K-theory}, (\href{http://arxiv.org/abs/math/0702145}{arXiv:math/0702145})

\end{itemize}
Specifically [[pullback]] and [[pushout]] of $C^\ast$-algebras is discussed in

\begin{itemize}%
\item Gerd Petersen, \emph{Pullback and pushout constructions in $C^\ast$-algebra theory} (\href{http://www.math.ru.nl/~mueger/ped2.pdf}{pdf})

\end{itemize}
The [[homotopy theory]] of $C^\ast$-algebras (a [[category of fibrant objects]]-structure on $C^\ast Alg$) is discussed in

\begin{itemize}%
\item Otgonbayar Uuye, \emph{Homotopy theory for $C^\ast$-algebras} (\href{http://arxiv.org/abs/1011.2926}{arXiv:1011.2926})

\end{itemize}
For more along such lines see the references at \emph{[[KK-theory]]} and \emph{[[E-theory]]}.

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