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Title: On the Lebesgue decomposition of the normal states of a JBW-algebra (English)
Author: Dubois, Jacques
Author: Hadjou, Brahim
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 117
Issue: 2
Year: 1992
Pages: 185-193
Summary lang: English
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Category: math
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Summary: In this article, a theorem is proved asserting that any linear functional defined on a JBW-algebra admits a Lebesque decomposition with respect to any normal state defined on the algebra. Then we show that the positivity (and the unicity) of this decomposition is insured for the trace states defined on the algebra. In fact, this property can be used to give a new characterization of the trace states amoungst all the normal states. (English)
Keyword: linear functional
Keyword: JBW-algebra
Keyword: Lebesgue decomposition
Keyword: normal state
Keyword: trace states
Keyword: state
Keyword: Lebesgue decomposition of a linear functional with respect to another linear functional
Keyword: support of linear functional
MSC: 06C15
MSC: 17C50
MSC: 46H70
MSC: 46L30
MSC: 46L50
MSC: 46L51
MSC: 46L53
MSC: 46L54
MSC: 46L70
MSC: 81B10
MSC: 81P10
idZBL: Zbl 0762.46061
idMR: MR1165895
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Date available: 2009-09-24T20:52:10Z
Last updated: 2015-08-30
Stable URL: http://hdl.handle.net/10338.dmlcz/125900
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Reference: [1] E. M. Alfsen, F. W. Shultz: On Non-commutative Spectral Theory and Jordan Algebras.Pгoc. London Math. Soc. 38 (1979), 497-516. Zbl 0404.46028, MR 0532984, 10.1112/plms/s3-38.3.497
Reference: [2] S. A. Ajupov: A New Proof of the Existence of Traces on Jordan Opeгator Algebras and Real von Newmann Algebras.J. of Functional Analysis 84 (1989), 312-321. MR 1001463, 10.1016/0022-1236(89)90100-6
Reference: [3] L. J. Bunce, J. D. Wright: Continuity and Linear Extensions of Quantum Measures on Jordan Operatoг Algebras.Preprint. To appear in Math. Scandinavia.
Reference: [4] A. M. Gleason: Measures on the closed subspaces of a Hilbeгt space.J. Math. Mech. 6 (1957), 885-893. MR 0096113
Reference: [5] H. Hanche-Olsen, E. Størmer: Jordan Operator Algebras.Pitman, Boston, 1984. MR 0755003
Reference: [6] V. Palko: On the Lebesgue Decomposition of Gleason Measures.Časopis pro pěstování Mat. 112 no. 1 (1987), 1-5. Zbl 0621.46058, MR 0880929
Reference: [7] G. K. Pederson, E. Størmer: Traces on Jordan Algebras.Can. J. Math. 34 (1982), 370-373. MR 0658972, 10.4153/CJM-1982-024-7
Reference: [8] G. T. Rüttimann, C. Schindler: The Lebesgue Decomposition of Measures on Orthomodular Posets.Quart. J. Math. Oxford 31 no. 2 (1986), 321-345. MR 0854631, 10.1093/qmath/37.3.321
Reference: [9] F. W. Shultz: On Normed Jordan Algebras Which Are Banach Dual Spaces.J. Funct. Analysis 31 (1979), 360-376. Zbl 0421.46043, MR 0531138, 10.1016/0022-1236(79)90010-7
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