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Keywords:
conditional expectation of an observable; partially compatible
conditional expectation of an observable; partially compatible
Summary:
In this paper, the authors introduce the notion of conditional expectation of an observable $x$ on a logic with respect to a sublogic, in a state $m$, relative to an element $a$ of the logic. This conditional expectation is an analogue of the expectation of an integrable function on a probability space.
References:
[1] V. S. Varadarajan: Geometry of Quantum Theory. Vol. 1, Princeton, New Yersey, Van Nostrand Reinhold 1968. MR 0471674 | Zbl 0155.56802
[2] S. Pulmannová: Compatibility and partial compatibility in quantum logics. Ann. Inst. H. Poincaré XXXIV (1981), 391-403. MR 0625170
[3] G. Brims G. Kalmbach: Some remarks on free orthomodular lattices. Proc. Univ. of Houston, Lattice Theory Conf. Houston (1973).
[4] N. Zierler: Axioms for non-relativistic quantum mechanics. Рас. J. Math. II, (1961), 1151-1169. MR 0140972 | Zbl 0138.44503
[5] S. P. Gudder J. Zerbe: Generalized monotone convergence and Randon-Nikodym theorems. J. Math Phys. 22 (1981) 2553-2561. DOI 10.1063/1.524832 | MR 0640666
[6] A. Dvurečenskij S. Pulmannová: On the sum of observables in a logic. Math. Slovaca, 30, (1980), 393-399. MR 0595300
[7] H. Mullikin S. P. Gudder: Measure theoretic convergences of observables and operators. J. Math. Phys. 14, (1973), 234-242. DOI 10.1063/1.1666301 | MR 0334747
[8] Z. Šidák: On relations between strict sense and wide sense conditional expectations. Teorija verojatnostej i jej o primenenija, II. 2., (1957), 283 - 288. MR 0092249
[9] S. P. Gudder: Uniqueness and existence properties of bounded observables. Pacific. J. Math. 15, (1966), 81-93, 588-589. MR 0201146 | Zbl 0149.23603
[10] P. Pták V. Rogalewicz: Regularly full logics and the uniqueness problem for observables. Ann. Inst. H. Poincaré 37, (1983), 69 - 74. MR 0700701
[11] Moy, Shu-Teh Chen: Characterization of conditional expectations as a transformation of function spaces. Pacif. J. Math. 4, (1954), 47-69. DOI 10.2140/pjm.1954.4.47 | MR 0060750
[12] R. R. Bahadur: Measurable subspaces and subalgebras. Proc. Amer. Math. Soc. 6, (1955) 565-570. DOI 10.1090/S0002-9939-1955-0072446-8 | MR 0072446 | Zbl 0066.11003
[13] S. P. Gudder: Joint distributions of observables. J. Math. Mech. 18, (1968), 269-302. MR 0232582 | Zbl 0241.60092
[14] J. M. Jauch: The quantum probability calculus. Synthese 29, (1974), 331 - 356. DOI 10.1007/BF00484955 | Zbl 0342.60004
[15] A. Dvurečenskij S. Pulmannová: Connection between joint distributions and compatibility. Rep. Math. Phys., 19, (1984), 349-359. DOI 10.1016/0034-4877(84)90007-7 | MR 0745430
[16] S. P. Gudder J. P. Marchand: Noncommutative probability on von Neumann algebras. J. Math. Phys. 13, (1972), 799-806. DOI 10.1063/1.1666054 | MR 0302108
[17] H. Umegaki: Conditional expectation in an operator algebra I. Tohoku Math. J. 6, (1954), 177-181. DOI 10.2748/tmj/1178245177 | MR 0068751
[18] M. Nakamura T. Turumaru: Expectations on an operator algebra. Tohoku Math. J. 6, (1954), 182-188. DOI 10.2748/tmj/1178245178 | MR 0068750
[19] M. Takesaki: Conditional expectations on von Neumann algebras. J. Funct. Anal. 9, (1972), 306-321. DOI 10.1016/0022-1236(72)90004-3 | MR 0303307
[20] H. Cycon K. E. Hellwig: Conditional expectations in generalized probability theory. J. Math. Phys. 18, (1977), 1154-1161. DOI 10.1063/1.523385 | MR 0436828
[21] S. P. Gudder J. P. Marchand: Conditional expectations on von Neumann algebras. A new approach. Rep. Math. Phys. 12, (1977), 317-329. DOI 10.1016/0034-4877(77)90030-1 | MR 0493386
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