# 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 $CQG_{lc}$ (http://planetmath.org/LocallyCompactQuantumGroup) by equipping such algebras with a co-associative multiplication (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, $JBWA$

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 the following.

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 (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 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.

## 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 21-23, 2002., Walter de Gruyter Gmbh & Co: Berlin.
• 2 Von Neumann and the http://plato.stanford.edu/entries/qt-nvd/Foundations of Quantum Theory.
• 3 B$\"{o}$hm, A., 1966, Rigged Hilbert Space and Mathematical Description of Physical Systems, Physica A, 236: 485-549.
• 4 B$\"{o}$hm, 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 Nucl$\'{e}$aires, 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 (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 LocallyCompactQuantumGroupsFromVonNeumannCAlgebrasWithHaarMeasures 2013-03-22 18:24:28 2013-03-22 18:24:28 bci1 (20947) bci1 (20947) 16 bci1 (20947) Topic msc 47A70 msc 46N50 msc 47L30 msc 47N50 msc 81P15 msc 46C05 locally compact quantum groups quantum groupoids Hopf and weak Hopf algebras HilbertSpace QuantumSpaceTimes VonNeumannAlgebra WeakHopfCAlgebra2 ClassificationOfHilbertSpaces QuantumSpaceTimes VonNeumannAlgebra WeakHopfCAlgebra2 JordanBanachAndJordanLieAlgebras QuantumLogic Distribution4 JBW-algebras $JBW$ algebras $JB$–algebra $JBWA$ $JL$ Jordan-Banach-von Neumann algebras $CQG_{lc}$