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 foundationPlanetmathPlanetmath for Quantum Mechanics in the form of W*-algebrasPlanetmathPlanetmathPlanetmath (http://planetmath.org/WeakHopfCAlgebra2) of (quantum) bounded operatorsMathworldPlanetmathPlanetmath in a (quantum:= presumed separable, i.e. with a countable basis) Hilbert spaceMathworldPlanetmath HS. Recently, such von Neumann algebrasMathworldPlanetmathPlanetmathPlanetmath, W* (http://planetmath.org/WeakHopfCAlgebra2) and/or (more generally) C*-algebras are, for example, employed to define locally compact quantum groupsPlanetmathPlanetmath C⁢Q⁢Gl⁢c (http://planetmath.org/LocallyCompactQuantumGroup) by equipping such algebras with a co-associative multiplicationPlanetmathPlanetmath (http://planetmath.org/WeakHopfCAlgebra2) and also with associated, both left– and right– Haar measures, defined by two semi-finite normal weights [1].

0.1.1 Remark on Jordan-Banach-von Neumann (JBW) algebras, J⁢B⁢W⁢A

A Jordan–Banach algebraMathworldPlanetmath (a JB–algebra for short) is both a real Jordan algebraPlanetmathPlanetmath and a Banach spaceMathworldPlanetmath, where for all S,T∈𝔄ℝ, we have the following.

A JLB–algebra is a J⁢B–algebra 𝔄ℝ together with a Poisson bracket for which it becomes a Jordan–Lie algebraMathworldPlanetmath J⁢L for some ℏ2≥0 . Such JLB–algebras often constitute the real part of several widely studied complex associative algebras. For the purpose of quantization, there are fundamental relationsMathworldPlanetmathPlanetmath between 𝔄s⁢a, JLB and Poisson algebras (http://planetmath.org/JordanBanachAndJordanLieAlgebras).

Definition 0.1.

A JB–algebra which is monotone completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and admits a separating set of normal sets is called a JBW-algebra.

These appeared in the work of von Neumann who developed an orthomodular lattice theory of projections on L⁢(H) on which to study quantum logicPlanetmathPlanetmath. BW-algebras have the following property: whereas 𝔄s⁢a is a J(L)B–algebra, the self-adjoint 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 GroupoidsPlanetmathPlanetmath”:
    Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002., Walter de Gruyter Gmbh & Co: Berlin.
  • 2 Von Neumann and the http://plato.stanford.edu/entries/qt-nvd/Foundations of Quantum TheoryPlanetmathPlanetmath.
  • 3 Böhm, A., 1966, Rigged Hilbert SpacePlanetmathPlanetmath and Mathematical Description of Physical Systems, Physica A, 236: 485-549.
  • 4 Böhm, A. and Gadella, M., 1989, Dirac Kets, Gamow Vectors and Gel’fand Triplets, New York: Springer-Verlag.
  • 5 Dixmier, J., 1981, Von Neumann Algebras, Amsterdam: North-Holland Publishing Company. [First published in French in 1957: Les Alge’bres d’Ope’rateurs dans l’Espace Hilbertien, Paris: Gauthier-Villars.]
  • 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]: 197-213. [Reprinted in C*-algebras: 1943-1993, in the series Contemporary Mathematics, 167, Providence, R.I. : American Mathematical Society, 1994.]
  • 7 Grothendieck, A., 1955, Produits Tensoriels Topologiques et Espaces Nucléaires, Memoirs of the American Mathematical Society, 16: 1-140.
  • 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 InfiniteMathworldPlanetmath Systems, first published in (Radei and Statzner 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.]
Title locally compact quantum groups from von Neumann/C*- algebras with Haar measures
Canonical name LocallyCompactQuantumGroupsFromVonNeumannCAlgebrasWithHaarMeasures
Date of creation 2013-03-22 18:24:28
Last modified on 2013-03-22 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 JBW-algebras
Defines J⁢B⁢W algebras
Defines J⁢B–algebra
Defines J⁢B⁢W⁢A
Defines J⁢L
Defines Jordan-Banach-von Neumann algebras
Defines C⁢Q⁢Gl⁢c