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quantum groups and von Neumann algebras (Topic)

Hilbert spaces, Von Neumann algebras and Quantum Groups

John von Neumann introduced a mathematical foundation for quantum mechanics in the form of $W^*$-algebras of (quantum) bounded operators in a (quantum:= presumed separable, i.e. with a countable basis) Hilbert space $H_S$. Recently, such von Neumann algebras, $W^*$ and/or (more generally) C*-algebras are, for example, employed to define locally compact quantum groups $CQG_{lc}$ by equipping such algebras with a co-associative multiplication and also with associated, both left– and right– Haar measures, defined by two semi-finite normal weights [1].

Remark on Jordan-Banach-von 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 $S, T \in \mathfrak{A}_{\mathbb{R}}$, we have

$\displaystyle \begin{aligned}\Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T \Vert^2 &\leq \Vert S^2 + T^2 \Vert ~. \end{aligned}$

A JLB–algebra is a $JB$–algebra $\mathfrak{A}_{\mathbb{R}}$ 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 complex associative algebras. For the purpose of quantization, there are fundamental relations between $\mathfrak{A}^{sa}$, JLB and Poisson algebras.

Definition 0.1   A JB–algebra which is monotone complete 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 $\mathcal L(H)$ on which to study quantum logic. 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.

Bibliography

1
Leonid Vainerman. 2003. “Locally Compact Quantum Groups and Groupoids”:
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 Foundations of Quantum Theory.
3
Böhm, A., 1966, Rigged Hilbert Space 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 Algèbres d'Opérateurs 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 Mathématique [Matematicheskii Sbornik] Nouvelle Série, 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 Infinite Systems, first published in (Rédei and Stöltzner 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.]



"quantum groups and von Neumann algebras" is owned by bci1.

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See Also: Hilbert space, topic on groupoid symmetries and duality, Hilbert space, quantum symmetry

Also defines:  W*-algebra, von Neumann algebra, JB-algebra, JBW-algebra, weak Hopf C*-algebra, Jordan-Banach-von Neumann algebra, $JBWA$, locally compact quantum group
Keywords:  quantum groups, von Neumann algebras

Cross-references: quantum logic, orthomodular lattice theory, work, relations, quantization, complex associative algebras, Banach space, Haar measures, C*-algebras, Hilbert space, operators, quantum mechanics
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This is version 7 of quantum groups and von Neumann algebras, born on 2009-02-18, modified 2009-05-21.
Object id is 539, canonical name is QuantumGroupsAndVonNeumannAlgebras.
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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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