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, suchvon Neumann algebras, $W^{*}$ (http://planetmath.org/WeakHopfCAlgebra2) and/or (more generally) C*-algebras are, for example, employed to definelocally compact quantum groups $CQG_{lc}$ (http://planetmath.org/LocallyCompactQuantumGroup) by equipping suchalgebras 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, $J B W A$

A Jordan–Banach algebra (a JB–algebra for short) is both a real Jordan algebra and aBanach space, where for all $S,T\\in\\mathfrak{A}_{\\mathbb{R}}$ , we have the following.

A JLB–algebra is a $J B$ –algebra $\\mathfrak{A}_{\\mathbb{R}}$ together with a Poisson bracket forwhich it becomes a Jordan–Lie algebra $J L$ for some $\\hslash^{2}\\geq 0$ . Such JLB–algebras oftenconstitute 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 iscalled 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 thehttp://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.]

单词	LocallyCompactQuantumGroupsFromVonNeumannCAlgebrasWithHaarMeasures
释义	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, suchvon Neumann algebras, $W^{}$ (http://planetmath.org/WeakHopfCAlgebra2) and/or (more generally) C-algebras are, for example, employed to definelocally compact quantum groups $CQG_{lc}$ (http://planetmath.org/LocallyCompactQuantumGroup) by equipping suchalgebras 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, $J B W A$ A Jordan–Banach algebra (a JB–algebra for short) is both a real Jordan algebra and aBanach space, where for all $S,T\\in\\mathfrak{A}_{\\mathbb{R}}$ , we have the following. A JLB–algebra is a $J B$ –algebra $\\mathfrak{A}_{\\mathbb{R}}$ together with a Poisson bracket forwhich it becomes a Jordan–Lie algebra $J L$ for some $\\hslash^{2}\\geq 0$ . Such JLB–algebras oftenconstitute 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 iscalled 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 thehttp://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.]
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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

0.1.1 Remark on Jordan-Banach-von Neumann (JBW) algebras, J⁢B⁢W⁢A

Definition 0.1.

References

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

0.1.1 Remark on Jordan-Banach-von Neumann (JBW) algebras, $J B W A$