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%%% Primary Title: quantum groups and von Neumann algebras
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\begin{document}
\subsection{Hilbert spaces, Von Neumann algebras and Quantum Groups}
John von Neumann introduced a mathematical foundation for \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} in the form of
\htmladdnormallink{$W^*$-algebras}{http://planetphysics.us/encyclopedia/WeakHopfCAlgebra.html}
of (quantum) bounded \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} in a (quantum:= presumed \emph{separable}, i.e. with a countable basis) \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $H_S$. Recently, such
\htmladdnormallink{von Neumann algebras, $W^*$}{http://planetphysics.us/encyclopedia/WeakHopfCAlgebra2.html} and/or (more generally) \htmladdnormallink{C*-algebras}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html} are, for example, employed to define
\htmladdnormallink{locally compact quantum groups $CQG_{lc}$}{http://planetphysics.us/encyclopedia/LocallyCompactQuantumGroup.html} by equipping such
\htmladdnormallink{algebras with a co-associative multiplication}{http://planetphysics.us/encyclopedia/WeakHopfCAlgebra2.html}
and also with associated, both left-- and right-- \htmladdnormallink{Haar measures}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html}, defined by two semi-finite normal weights
\cite{Vainerman2003}.
\subsubsection{Remark on Jordan-Banach-von Neumann (JBW) algebras, $JBWA$}
A \emph{Jordan--Banach algebra} (a JB--algebra for short) is both a real Jordan algebra and a
\htmladdnormallink{Banach space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html}, where for all $S, T \in \mathfrak A_{\bR}$, we have
\bigbreak
\bigbreak
\bigbreak
$$ \begin{aligned} \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T
\Vert^2 &\leq \Vert S^2 + T^2 \Vert ~. \end{aligned}$$
\bigbreak
\bigbreak
\bigbreak
A \emph{JLB--algebra} is a $JB$--algebra $\mathfrak A_{\bR}$ together with a Poisson bracket for
which it becomes a Jordan--Lie algebra $JL$ for some $\hslash^2 \geq 0$~. Such JLB--algebras often
constitute the real part of several widely studied \htmladdnormallink{complex associative algebras}{http://planetphysics.us/encyclopedia/OrthomodularLatticeTheory.html}.
For the purpose of \htmladdnormallink{quantization}{http://planetphysics.us/encyclopedia/MoyalDeformation.html}, there are fundamental \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html} between
\htmladdnormallink{$\mathfrak A^{sa}$, JLB and Poisson algebras}{http://planetphysics.us/encyclopedia/JordanBanachAndJordanLieAlgebras.html}.
\bigbreak
\begin{definition}
A JB--algebra which is monotone complete and admits a separating set of normal sets is
called a \emph{JBW-algebra}.
\end{definition}
These appeared in the \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} of von Neumann who developed an \emph{\htmladdnormallink{orthomodular lattice theory}{http://planetphysics.us/encyclopedia/OrthomodularLatticeTheory.html} of projections on $\mathcal L(H)$} on which to study \emph{\htmladdnormallink{quantum logic}{http://planetphysics.us/encyclopedia/LQG2.html}}. BW-algebras have the following property: whereas $\mathfrak A^{sa}$ is a J(L)B--algebra, the self-adjoint part of a von Neumann algebra is a JBW--algebra.
\begin{thebibliography}{9}
\bibitem{Vainerman2003}
Leonid Vainerman. 2003.
\htmladdnormallink{``Locally Compact Quantum Groups and Groupoids'': \\
Proceedings of the Meeting of Theoretical Physicists and Mathematicians}{http://planetmath.org/?op=getobj&from=books&id=160}, Strasbourg, February 21-23, 2002., Walter de Gruyter Gmbh \& Co: Berlin.
\bibitem{QTF}
Von Neumann and the
\htmladdnormallink{Foundations of Quantum Theory.}{http://plato.stanford.edu/entries/qt-nvd/}
\bibitem{Bohm66}
B\"ohm, A., 1966, Rigged Hilbert Space and Mathematical Description of Physical Systems, {\em Physica A}, 236: 485-549.
\bibitem{Bohm89}
B\"ohm, A. and Gadella, M., 1989, \emph{Dirac Kets, Gamow Vectors and Gel'fand Triplets}, New York: Springer-Verlag.
\bibitem{DJ81}
Dixmier, J., 1981, \emph{Von Neumann Algebras}, Amsterdam: North-Holland Publishing Company. [First published in French in 1957: \emph{Les Alg\`ebres d'Op\'erateurs dans l'Espace Hilbertien}, Paris: Gauthier-Villars.]
\bibitem{GINM43}
Gelfand, I. and Neumark, M., 1943, On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space,
{\em Recueil Math\'ematique} [Matematicheskii Sbornik] Nouvelle S\'erie, 12 [54]: 197-213. [Reprinted in C*-algebras: 1943-1993, in the series Contemporary Mathematics, 167, Providence, R.I. : American Mathematical Society, 1994.]
\bibitem{Alex55}
Grothendieck, A., 1955, Produits Tensoriels Topologiques et Espaces Nucl\'eaires,
\emph{Memoirs of the American Mathematical Society}, 16: 1-140.
\bibitem{HS90}
Horuzhy, S. S., 1990, {\em Introduction to Algebraic Quantum Field Theory}, Dordrecht: Kluwer Academic Publishers.
\bibitem{JV55}
J. von Neumann.,1955, {\em Mathematical Foundations of Quantum Mechanics.}, Princeton, NJ: Princeton University Press. [First published in German in 1932: {\em Mathematische Grundlagen der Quantenmechanik}, Berlin: Springer.]
\bibitem{JV37}
J. von Neumann, 1937, {\em Quantum Mechanics of Infinite Systems}, first published in (R\'edei and St\"oltzner 2001, 249-268). [A mimeographed version of a lecture given at Pauli's seminar held at the Institute for Advanced Study in 1937, John von Neumann Archive, Library of Congress, Washington, D.C.]
\end{thebibliography}
\end{document}
