PlanetMath: quantum groups and von Neumann algebras

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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

-algebras of (quantum) bounded operators in a (quantum:= presumed separable, i.e. with a countable basis) Hilbert space

. Recently, such von Neumann algebras,

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,

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 -algebra $\mathfrak{A}_{\mathbb{R}}$ together with a Poisson bracket for which it becomes a Jordan-Lie algebra 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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Other names:

locally compact quantum groups, quantum groupoids, Hopf and weak Hopf algebras

Also defines:

JBW-algebras, $JBW$

algebras,

--algebra, $JBWA$

, Jordan-Banach-von Neumann algebras, $CQG_{lc}$

Keywords:

quantum groupoids, Hopf and weak Hopf algebras

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Cross-references: von Neumann algebra, self-adjoint, property, quantum logic, projections, theory, orthomodular lattice, separating, complete, monotone, relations, quantization, associative, complex, real part, Poisson bracket, Banach space, Jordan algebra, real, algebra, weights, normal, Haar measures, C*-algebras, Hilbert space, countable basis, separable, bounded operators, mathematical foundation, John von Neumann
There are 14 references to this entry.

This is version 5 of quantum groups and von Neumann algebras, born on 2008-09-21, modified 2008-10-17.
Object id is 11056, canonical name is QuantumGroupsAndVonNeumannAlgebras.
Accessed 847 times total.

Classification:

AMS MSC:	46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )
	81P15 (Quantum theory :: Axiomatics, foundations, philosophy :: Quantum measurement theory)
	47N50 (Operator theory :: Miscellaneous applications of operator theory :: Applications in quantum physics)
	47L30 (Operator theory :: Linear spaces and algebras of operators :: Abstract operator algebras on Hilbert spaces)
	46N50 (Functional analysis :: Miscellaneous applications of functional analysis :: Applications in quantum physics)
	47A70 (Operator theory :: General theory of linear operators :: eigenfunction expansions; rigged Hilbert spaces)

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