locally compact quantum groups from von Neumann/${C}^{*}$ algebras with Haar measures
0.1 Hilbert spaces, Von Neumann algebras and Quantum Groups
John von Neumann introduced a mathematical foundation^{} for Quantum Mechanics in the form of ${W}^{*}$algebras^{} (http://planetmath.org/WeakHopfCAlgebra2) of (quantum) bounded operators^{} in a (quantum:= presumed separable, i.e. with a countable basis) Hilbert space^{} ${H}_{S}$. Recently, such von Neumann algebras^{}, ${W}^{*}$ (http://planetmath.org/WeakHopfCAlgebra2) and/or (more generally) C*algebras are, for example, employed to define locally compact quantum groups^{} $CQ{G}_{lc}$ (http://planetmath.org/LocallyCompactQuantumGroup) by equipping such algebras with a coassociative multiplication^{} (http://planetmath.org/WeakHopfCAlgebra2) and also with associated, both left– and right– Haar measures, defined by two semifinite normal weights [1].
0.1.1 Remark on JordanBanachvon Neumann (JBW) algebras, $JBWA$
A Jordan–Banach algebra^{} (a JB–algebra for short) is both a real Jordan algebra^{} and a Banach space^{}, where for all $S,T\in {\U0001d504}_{\mathbb{R}}$, we have the following.
A JLB–algebra is a $JB$–algebra ${\U0001d504}_{\mathbb{R}}$ together with a Poisson bracket for which it becomes a Jordan–Lie algebra^{} $JL$ for some ${\mathrm{\hslash}}^{2}\ge 0$ . Such JLB–algebras often constitute the real part of several widely studied complex associative algebras. For the purpose of quantization, there are fundamental relations^{} between ${\U0001d504}^{sa}$, JLB and Poisson algebras (http://planetmath.org/JordanBanachAndJordanLieAlgebras).
Definition 0.1.
A JB–algebra which is monotone complete^{} and admits a separating set of normal sets is called a JBWalgebra.
These appeared in the work of von Neumann who developed an orthomodular lattice theory of projections on $\mathrm{L}\mathit{}\mathrm{(}H\mathrm{)}$ on which to study quantum logic^{}. BWalgebras have the following property: whereas ${\U0001d504}^{sa}$ is a J(L)B–algebra, the selfadjoint part of a von Neumann algebra is a JBW–algebra.
References

1
Leonid Vainerman. 2003.
http://planetmath.org/?op=getobj&from=books&id=160“Locally Compact Quantum Groups and Groupoids^{}”:
Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 2123, 2002., Walter de Gruyter Gmbh & Co: Berlin.  2 Von Neumann and the http://plato.stanford.edu/entries/qtnvd/Foundations of Quantum Theory^{}.
 3 B$\mathrm{\xf6}$hm, A., 1966, Rigged Hilbert Space^{} and Mathematical Description of Physical Systems, Physica A, 236: 485549.
 4 B$\mathrm{\xf6}$hm, A. and Gadella, M., 1989, Dirac Kets, Gamow Vectors and Gel’fand Triplets, New York: SpringerVerlag.
 5 Dixmier, J., 1981, Von Neumann Algebras, Amsterdam: NorthHolland Publishing Company. [First published in French in 1957: Les Alge’bres d’Ope’rateurs dans l’Espace Hilbertien, Paris: GauthierVillars.]
 6 Gelfand, I. and Neumark, M., 1943, On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space, Recueil Mathe’matique [Matematicheskii Sbornik] Nouvelle Se’rie, 12 [54]: 197213. [Reprinted in C*algebras: 19431993, in the series Contemporary Mathematics, 167, Providence, R.I. : American Mathematical Society, 1994.]
 7 Grothendieck, A., 1955, Produits Tensoriels Topologiques et Espaces Nucl$\mathrm{\xe9}$aires, Memoirs of the American Mathematical Society, 16: 1140.
 8 Horuzhy, S. S., 1990, Introduction to Algebraic Quantum Field Theory, Dordrecht: Kluwer Academic Publishers.
 9 J. von Neumann.,1955, Mathematical Foundations of Quantum Mechanics., Princeton, NJ: Princeton University Press. [First published in German in 1932: Mathematische Grundlagen der Quantenmechanik, Berlin: Springer.]
 10 J. von Neumann, 1937, Quantum Mechanics of Infinite^{} Systems, first published in (Radei and Statzner 2001, 249268). [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.]
Title  locally compact quantum groups from von Neumann/${C}^{*}$ algebras with Haar measures 

Canonical name  LocallyCompactQuantumGroupsFromVonNeumannCAlgebrasWithHaarMeasures 
Date of creation  20130322 18:24:28 
Last modified on  20130322 18:24:28 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  16 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 47A70 
Classification  msc 46N50 
Classification  msc 47L30 
Classification  msc 47N50 
Classification  msc 81P15 
Classification  msc 46C05 
Synonym  locally compact quantum groups 
Synonym  quantum groupoids 
Synonym  Hopf and weak Hopf algebras 
Related topic  HilbertSpace 
Related topic  QuantumSpaceTimes 
Related topic  VonNeumannAlgebra 
Related topic  WeakHopfCAlgebra2 
Related topic  ClassificationOfHilbertSpaces 
Related topic  QuantumSpaceTimes 
Related topic  VonNeumannAlgebra 
Related topic  WeakHopfCAlgebra2 
Related topic  JordanBanachAndJordanLieAlgebras 
Related topic  QuantumLogic 
Related topic  Distribution4 
Defines  JBWalgebras 
Defines  $JBW$ algebras 
Defines  $JB$–algebra 
Defines  $JBWA$ 
Defines  $JL$ 
Defines  JordanBanachvon Neumann algebras 
Defines  $CQ{G}_{lc}$ 