# PlanetPhysics/Quantum Groups and Von Neumann Algebras

\newcommand{\sqdiagram}[9]{$\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}$ }

### Hilbert spaces, Von Neumann algebras and Quantum Groups

John von Neumann introduced a mathematical foundation for quantum mechanics in the form of ${\displaystyle W^{*}}$-algebras of (quantum) bounded operators in a (quantum:= presumed separable , i.e. with a countable basis) Hilbert space ${\displaystyle H_{S}}$. Recently, such von Neumann algebras, ${\displaystyle W^{*}}$ and/or (more generally) C*-algebras are, for example, employed to define \htmladdnormallink{locally compact quantum groups ${\displaystyle CQG_{lc}}$}{http://planetphysics.us/encyclopedia/LocallyCompactQuantumGroup.html} 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, ${\displaystyle JBWA}$

A Jordan--Banach algebra (a JB--algebra for short) is both a real Jordan algebra and a Banach space, where for all $\displaystyle S, T \in \mathfrak A_{\bR}$ , we have \bigbreak \bigbreak \bigbreak $\displaystyle \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T \Vert^2 &\leq \Vert S^2 + T^2 \Vert ~.$ \bigbreak \bigbreak \bigbreak

A JLB--algebra is a ${\displaystyle JB}$--algebra $\displaystyle \mathfrak A_{\bR}$ together with a Poisson bracket for which it becomes a Jordan--Lie algebra ${\displaystyle JL}$ for some ${\displaystyle \hslash ^{2}\geq 0}$~. 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 \htmladdnormallink{${\displaystyle {\mathfrak {A}}^{sa}}$, JLB and Poisson algebras}{http://planetphysics.us/encyclopedia/JordanBanachAndJordanLieAlgebras.html}. \bigbreak 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 \htmladdnormallink{orthomodular lattice theory {http://planetphysics.us/encyclopedia/OrthomodularLatticeTheory.html} of projections on ${\displaystyle {\mathcal {L}}(H)}$} on which to study quantum logic. BW-algebras have the following property: whereas ${\displaystyle {\mathfrak {A}}^{sa}}$ is a J(L)B--algebra, the self-adjoint part of a von Neumann algebra is a JBW--algebra.

## References

1. Leonid Vainerman. 2003. \htmladdnormallink{"Locally Compact Quantum Groups and Groupoids": \\ Proceedings of the Meeting of Theoretical Physicists and Mathematicians}{http://planetmath.org/?op=getobj&from=books&id=160}, Strasbourg, February 21-23, 2002., Walter de Gruyter Gmbh \& Co: Berlin.
2. Von Neumann and the Foundations of Quantum Theory.
3. B\"ohm, A., 1966, Rigged Hilbert Space and Mathematical Description of Physical Systems, Physica A , 236: 485-549.
4. B\"ohm, 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 Alg\`ebres d'Op\'erateurs 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 Math\'ematique [Matematicheskii Sbornik] Nouvelle S\'erie, 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\'eaires, 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 (R\'edei and St\"oltzner 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.]