quantum groups and von Neumann algebras
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
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].
A Jordan-Banach algebra (a JB-algebra for short) is both a real Jordan algebra and a
Banach space, where for all
, we have
A JLB-algebra is a
-algebra
together with a Poisson bracket for
which it becomes a Jordan-Lie algebra
for some
. 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
, 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
on which to study quantum logic. BW-algebras have the following property: whereas
is a J(L)B-algebra, the self-adjoint part of a von Neumann algebra is a JBW-algebra.
-
- 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.]
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by bci1.