CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE Lp-SPACES ASSOCIATED WITH CENTER-VALUED TRACES

doi:10.1016/S0252-9602(17)30055-3

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (4): 1019-1032.doi: 10.1016/S0252-9602(17)30055-3

CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE L_p-SPACES ASSOCIATED WITH CENTER-VALUED TRACES

Inomjon GANIEV¹, Farrukh MUKHAMEDOV²

1. Department of Science in Engineering, Faculty of Engineering, International Islamic University Malaysia, P. O. Box 10, 50728 Kuala-Lumpur, Malaysia;
2. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, 15551, Abu Dhabi, UAE

收稿日期:2016-03-29 修回日期:2016-07-04 出版日期:2017-08-25 发布日期:2017-08-25
作者简介:Inomjon GANIEV,E-mail:inam@iium.edu.my;Farrukh MUKHAMEDOV,E-mail:far75m@yandex.ru,farrukh m@uaeu.ac.ae
基金资助:
Supported by the MOHE Grant FRGS13-071-0312.

CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE L_p-SPACES ASSOCIATED WITH CENTER-VALUED TRACES

Inomjon GANIEV¹, Farrukh MUKHAMEDOV²

1. Department of Science in Engineering, Faculty of Engineering, International Islamic University Malaysia, P. O. Box 10, 50728 Kuala-Lumpur, Malaysia;
2. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, 15551, Abu Dhabi, UAE

Received:2016-03-29 Revised:2016-07-04 Online:2017-08-25 Published:2017-08-25
About author:Inomjon GANIEV,E-mail:inam@iium.edu.my;Farrukh MUKHAMEDOV,E-mail:far75m@yandex.ru,farrukh m@uaeu.ac.ae
Supported by:
Supported by the MOHE Grant FRGS13-071-0312.

摘要/Abstract

摘要：

In this paper we prove the existence of conditional expectations in the noncommutative L_p(M, Φ)-spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of weighted averages of martingales in noncommutative L_p(M, Φ)-spaces.

关键词: noncommutative L_p(M, Φ)-space, Banach-Kantorovich space, measurable, conditional expectation, measurable bundle, martingale

Abstract:

Key words: noncommutative L_p(M, Φ)-space, Banach-Kantorovich space, measurable, conditional expectation, measurable bundle, martingale

Inomjon GANIEV, Farrukh MUKHAMEDOV. CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE L_p-SPACES ASSOCIATED WITH CENTER-VALUED TRACES[J]. 数学物理学报(英文版), 2017, 37(4): 1019-1032.

Inomjon GANIEV, Farrukh MUKHAMEDOV. CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE L_p-SPACES ASSOCIATED WITH CENTER-VALUED TRACES[J]. Acta mathematica scientia,Series B, 2017, 37(4): 1019-1032.

参考文献

[1] Cecchini C, Petz D. Norm convergence of generalized martingales in L^p-spaces over von Neumann algebras. Acta Sci Math (Szeged), 1985, 48:55-63
[2] Cuculescu I. Martingales on von Neumann algebras. J Multivariate Anal, 1971, 1:17-27
[3] Chilin V I, Katz A A. On abstract characterization of non-commutative L_p-spaces associated with centervalued trace. Methods Funct Anal Topol, 2005, 11:346-355
[4] Chilin V I, Zakirov B S. Non-commutative L_p-spaces associated with a Maharam trace. J Operator Theory, 2012, 68:67-83
[5] Chilin V I, Zakirov B S. Maharam traces on von Neumann algebras. Methods Funct Anal Topol, 2010, 16(2):101-111
[6] Dang-Ngoc N. Pointwise convergence of martingales in von Neumann algebras. Israel J Math, 1979, 34:273-280
[7] Dixmier J. Les C_*-algebres et leurs representations. Paris, Gauthier-Villars Editeur, 1969
[8] Ganiev I G. The martingales convergence in the Banach-Kantorovich's lattices L_p(▽, μ). Uzb Math J, 2000, (1):18-26
[9] Ganiev I G, Chilin V I. Measurable bundles of noncommutative L^p-spaces associated with a center-valued trace. Siberian Adv Math, 2002, 12:19-33
[10] Ganiev I G, Chilin V I. Measurable bundles of C^*-algebras. Vladikavkaz Mat Zh, 2003, 5(1):35-38(Russian)
[11] Ganiev I G, Mukhamedov F. On the "Zero-Two" law for positive contractions in the Banach-Kantorovich lattice L^p(▽, μ). Comment Math Univ Carolinae, 2006, 47:427-436
[12] Ganiev I, Mukhamedov F. On weighted ergodic theorems for Banach-Kantorovich lattice L_p(▽, μ). Lobachevskii J Math, 34(2013), 1-10
[13] Ganiev I, Mukhamedov F. Measurable bundles of C^*-dynamical systems and its applications. Positivity, 2014, 18:687-702
[14] Ganiev I, Mukhamedov F. Weighted ergodic theorem for contractions of Orlicz-Kantorovich lattice L_M(▽b, μb). Bull Malays Math Sci Soc, 2015, 38:387-397
[15] Ganiev I, Mukhamedov F, Bekbaev D. The strong "zero-two" law for positive contractions of BanachKantorovich L_p-lattices. Turk J Math, 2015, 39:583-594
[16] Goldstein S. Norm convergence of martingales in L^p-spaces over von Neumann algebras. Rev Roumanie Math Pures Appl, 1987, 32:531-541
[17] Gutman A E. Banach bundles in the theory of lattice-normed spaces. I. Continuous Banach bundles. Siberian Adv Math, 1993, 3:1-55
[18] Gutman A E. Banach bundles in the theory of lattice-normed spaces, Ⅲ. Siberien Adv Math, 1993, 3:8-40
[19] Gutman A E. Banach fiberings in the theory of lattice-normed spaces. Order-compatible linear operators. Trudy Inst Mat, 1995, 29:63-211(Russian)
[20] Hiai F, Tsukada M. Generalized conditional expectations and martingales in noncommutative L^p-spaces. J Operator Theory, 1987, 18:265-288
[21] Junge M. Doob's inequality for non-commutative martingales. J Reine Angew Math, 2002, 549:149-190
[22] Karimov A, Mukhamedov F. On martingal convergence theorems in JW-algebras. Bull Malays Math Sci Soc, 2010, 33:405-410
[23] Kikuchi M. Convergence of weighted averages of martingales in Banach function spaces. J Math Anal Appl, 2000, 244:39-56
[24] Kusraev A G. Vector Duality and Its Applications. Nauka:Novosibirsk, 1985(in Russian)
[25] Kusraev A G. Linear operators in lattice-normed spaces//Studies on Geometry in the Large and Mathematical Analysis, Vol 9. Nauka:Novosibirsk, 1987(in Russian)
[26] Kusraev A G. Dominated Operators. Dordrecht:Kluwer Academic Publishers, 2000
[27] Lance E C. Martingale convergence in Neumann algebras. Math Proc Cambridge Philos Soc, 1978, 84:47-56
[28] Meyer P A. Quantum probability for Probabilists. Lecture Notes in Mathematics 1538. Berlin:Springer, 1993
[29] Muratov M A, Chilin V I. Algebras of measurable and locally measurable operators. Kyiv, Pratsi In-ty Matematiki NAN Ukraini, 2007(Russian)
[30] Nelson E. Note on non-commutative integration. J Funct Anal, 1974, 15:103-117
[31] Segal I E. A non-commutative extansion of abstract integration. Ann Math, 1953, 57:401-457
[32] Takesaki M. Conditional expectations in von Neumann algebras. J Funct Anal, 1972, 9:306-321
[33] Tsukada M. Strong convergence of martingales in von Neumann algebras. Proc Amer Math Soc, 1983, 88:537-540
[34] Umegaki H. Conditional expectation in an operator algebra. Tohoku Math J, 1954, 6:177-181
[35] Umegaki H. Conditional expectation in an operator algebra Ⅱ. Tohoku Math J, 1956, 8:86-100
[36] Xu Q. Operator spaces and noncommutative L_p-spaces. The part on noncommutative L_p-spaces. Lectures in the Summer School on Banach spaces and Operator spaces Nankai University China July 16-July 20, 2007
[37] Yeadon F J. Non-commutative L_p-spaces. Math Proc Camb Phil Soc, 1975, 77:91-102
[38] Zhang Ch, Hou Y. Convergence of weighed averages of martingales in noncommutative Banach function spaces. Acta Math Sci, 2012, 32B:735-744
[39] Zhang Ch, Hou Y. Convergence of weighted averages of noncommutative martingales. Sci China Math, 2013, 56:823-830

CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE L_p-SPACES ASSOCIATED WITH CENTER-VALUED TRACES

CONDITIONAL EXPECTATIONS AND MARTINGALES IN NONCOMMUTATIVE L_p-SPACES ASSOCIATED WITH CENTER-VALUED TRACES

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