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Homelocally compact quantum groups from von Neumann/$C^*$- algebras with Haar measures

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# 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
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].

# 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.

###### 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.
“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 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.]

## Mathematics Subject Classification

47A70*no label found*46N50

*no label found*47L30

*no label found*47N50

*no label found*81P15

*no label found*46C05

*no label found*

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