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\begin{document}
\subsection{C*- and von Neumann algebras: Quantum operator algebra in quantum theories}
\subsubsection{Introduction}
C*-algebra has evolved as a key \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} in Quantum Operator Algebra after the introduction of the von Neumann algebra for the mathematical foundation of \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html}. The von Neumann algebra \htmladdnormallink{classification}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} is simpler and studied in greater depth than that of general C*-algebra classification theory.
The importance of C*-algebras for understanding the geometry of \htmladdnormallink{quantum state spaces}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} (Alfsen and Schultz, 2003 \cite{AS}) cannot be overestimated. The theory of C*-algebras has numerous applications in the theory of \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and symmetric algebras, the theory of \htmladdnormallink{dynamical systems}{http://planetphysics.us/encyclopedia/ContinuousGroupoidHomomorphism.html}, statistical physics and \htmladdnormallink{quantum field theory}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html}, and also in the theory of \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} on a \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html}.
Moreover, the introduction of \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} C*-algebras in \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} has already played important roles in expanding the Hilbert space perspective of Quantum Mechanics developed by von Neumann. Furthermore, \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html} are currently being approached in terms of groupoid C*- \htmladdnormallink{convolution}{http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} algebra and their representations; the latter also enter into the construction of \htmladdnormallink{compact quantum groupoids}{http://planetphysics.us/encyclopedia/QuantumCompactGroupoids.html} as developed in the Bibliography cited, and also briefly outlined here in the second \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}.
The fundamental connections that exist between \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} of $C^*$-algebras and those of von Neumann and other \htmladdnormallink{quantum operator algebras}{http://planetphysics.us/encyclopedia/Groupoid.html}, such as JB- or JBL- algebras are yet to be completed and are the subject of in depth studies \cite{AS}.
\subsection{Basic definitions}
A \textbf{C*-algebra} is simultaneously a $*$--algebra and a \htmladdnormallink{Banach space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} -with additional conditions- as defined next.
Let us consider first the definition of an \emph{involution} on a complex algebra $\mathfrak A$.
\begin{definition}
An \emph{involution} on a complex algebra $\mathfrak A$ is a \emph{real--linear map} $T \mapsto T^*$
such that for all
$S, T \in \mathfrak A$ and $\lambda \in \bC$, we have $ T^{**} = T~,~ (ST)^* = T^* S^*~,~ (\lambda T)^* = \bar{\lambda} T^*~. $
\end{definition}
A \emph{*-algebra} is said to be a \htmladdnormallink{complex associative algebra}{http://planetphysics.us/encyclopedia/OrthomodularLatticeTheory.html} together with an \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} of involution $*$~.
\subsection{C*-algebra}
\begin{definition}
A \emph{C*-algebra} is simultaneously a *-algebra and a Banach space $\mathfrak A$,
satisfying for all $S, T \in \mathfrak A$~ the following conditions:
$ \begin{aligned} \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. \end{aligned}$
\end{definition}
One can easily verify that $\Vert A^* \Vert = \Vert A \Vert$~.
By the above axioms a C*--algebra is a special case of a Banach algebra where the latter requires the above C*-norm property, but not the involution (*) property.
Given Banach spaces $E, F$ the space $\mathcal L(E, F)$ of (bounded) \htmladdnormallink{linear operators}{http://planetphysics.us/encyclopedia/Commutator.html} from $E$ to $F$ forms a Banach space, where for $E=F$, the space $\mathcal L(E) = \mathcal L(E, E)$ is a Banach algebra with respect to the \htmladdnormallink{norm}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} \bigbreak
$\Vert T \Vert := \sup\{ \Vert Tu \Vert : u \in E~,~ \Vert u \Vert= 1 \}~. $
\bigbreak
In quantum field theory one may start with a Hilbert space $H$, and consider the Banach
algebra of bounded linear operators $\mathcal L(H)$ which given to be closed under the usual
\htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} operations and taking adjoints, forms a $*$--algebra of bounded operators, where the
adjoint operation \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} as the involution, and for $T \in \mathcal L(H)$ we have~:
$ \Vert T \Vert := \sup\{ ( Tu , Tu): u \in H~,~ (u,u) = 1 \}~,$ and $ \Vert Tu \Vert^2 = (Tu,
Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.$
By a \emph{\htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} between C*-algebras} $\mathfrak A,\mathfrak B$ we mean a linear map $\phi :
\mathfrak A \lra \mathfrak B$, such that for all $S, T \in \mathfrak A$, the following hold~:
\bigbreak
$\phi(ST) = \phi(S) \phi(T)~,~ \phi(T^*) = \phi(T)^*~, $
\bigbreak
where a \htmladdnormallink{bijective}{http://planetphysics.us/encyclopedia/Bijective.html} morphism is said to be an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} (in which case it is then an
isometry). A fundamental \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} is that any norm-closed $*$-algebra $\mathcal A$ in
$\mathcal L(H)$ is a C*-algebra, and conversely, any C*-algebra is isomorphic to a norm--closed $*$-algebra in $\mathcal L(H)$ for some Hilbert space $H$~.
One can thus also define \emph{the category $\mathcal{C}^*$ of C*-algebras and morphisms between C*-algebras}.
For a C*-algebra $\mathfrak A$, we say that $T \in \mathfrak A$ is \emph{self--adjoint} if $T
= T^*$~. Accordingly, the self--adjoint part $\mathfrak A^{sa}$ of $\mathfrak A$ is a real
\htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} since we can decompose $T \in \mathfrak A^{sa}$ as ~:
$ T = T' + T^{''} := \frac{1}{2} (T + T^*) + \iota (\frac{-\iota}{2})(T - T^*)~.$
A \emph{commutative} C*--algebra is one for which the associative multiplication is
commutative. Given a \htmladdnormallink{commutative C*--algebra}{http://planetphysics.us/encyclopedia/OrthomodularLatticeTheory.html} $\mathfrak A$, we have $\mathfrak A \cong C(Y)$,
the algebra of continuous functions on a compact Hausdorff space $Y~$.
The classification of {$C^*$-algebras} is far more complex than that of von Neumann algebras that provide
the fundamental algebraic content of quantum state and \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} spaces in quantum theories.
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\end{document}
