Geodesic
On the Lebesgue decomposition of the normal states of a JBW-algebra
Mathematica Bohemica, Tome 117 (1992) no. 2, pp. 185-193

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MR Zbl
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.
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.
DOI : 10.21136/MB.1992.125900
Classification : 06C15, 17C50, 46H70, 46L30, 46L50, 46L51, 46L53, 46L54, 46L70, 81B10, 81P10
Keywords: linear functional; JBW-algebra; Lebesgue decomposition; normal state; trace states; state; Lebesgue decomposition of a linear functional with respect to another linear functional; support of linear functional 
Dubois, Jacques; Hadjou, Brahim. On the Lebesgue decomposition of the normal states of a JBW-algebra. Mathematica Bohemica, Tome 117 (1992) no. 2, pp. 185-193. doi: 10.21136/MB.1992.125900
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[1] E. M. Alfsen, F. W. Shultz: On Non-commutative Spectral Theory and Jordan Algebras. Pгoc. London Math. Soc. 38 (1979), 497-516. | DOI | MR | Zbl

[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. | DOI | MR

[3] L. J. Bunce, J. D. Wright: Continuity and Linear Extensions of Quantum Measures on Jordan Operatoг Algebras. Preprint. To appear in Math. Scandinavia.

[4] A. M. Gleason: Measures on the closed subspaces of a Hilbeгt space. J. Math. Mech. 6 (1957), 885-893. | MR

[5] H. Hanche-Olsen, E. Størmer: Jordan Operator Algebras. Pitman, Boston, 1984. | MR

[6] V. Palko: On the Lebesgue Decomposition of Gleason Measures. Časopis pro pěstování Mat. 112 no. 1 (1987), 1-5. | MR | Zbl

[7] G. K. Pederson, E. Størmer: Traces on Jordan Algebras. Can. J. Math. 34 (1982), 370-373. | DOI | MR

[8] G. T. Rüttimann, C. Schindler: The Lebesgue Decomposition of Measures on Orthomodular Posets. Quart. J. Math. Oxford 31 no. 2 (1986), 321-345. | DOI | MR

[9] F. W. Shultz: On Normed Jordan Algebras Which Are Banach Dual Spaces. J. Funct. Analysis 31 (1979), 360-376. | DOI | MR | Zbl

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