On non-commutative ergodic type theorems for free finitely generated semigroups
Author: Grabarnik Genady Ya., Katz Alexander A., Shwartz Larisa A.
Journal: Владикавказский математический журнал @vmj-ru
Article in issue: 1 т.9, 2007.
Free access
In the paper the authors generalize Bufetov's Ergodic Type Theorems to the case of the actions of free finitely generated semigroups on von Neumann algebras.
Short address: https://sciup.org/14318204
IDR: 14318204 | UDC: 517.98
References On non-commutative ergodic type theorems for free finitely generated semigroups
- Akcoglu M. A. A pointwise ergodic theorem in L_p-spaces//Canad. J. Math.-1975.-V. 27, № 5.-P. 1075-1082.
- Bufetov A. I. Ergodic theorems for actions of several mappings//Uspekhi Math. Nauk.-1999.-V. 54, № 4 (328).-P. 159-160; Translation in Russian Math. Surveys.-1999.-V. 54, № 4.-P. 835-836.
- Bufetov A. I. Operator ergodic theorems for actions of free semigroups and groups//Funktsional. Anal. i Prilozhen.-2000.-V. 34, № 4.-P. 1-17; translation in Funct. Anal. Appl.-2000.-V. 34, № 4.-P. 239-251.
- Dixmier J. Les algebres doperateurs dans lespace hilbertien (algebres de von Neumann). Reprint of the second (1969) edition. Les Grands Classiques Gauthier-Villars.-Paris: Editions Jacques Gabay, 1996.-367 p.-In French.
- Goldstein M. Sh. Theorems of almost everywhere convergence in von Neumann algebras//J. Oper. Theory.-1981.-V. 6.-P. 233-311.-In Russian.
- Goldstein M. Sh., Grabarnik G. Ya. Almost sure convergence theorems in von Neumann algebras//Israel J. Math.-1991.-V. 76, № 1/2.-P. 161-182.
- Grabarnik G. Ya., Katz A. A., Shwartz L. Ergodic type theorems for actions of finitely generated semigroups on von neumann algebras. I//Proceedings of the 3rd Annual Hawaii International Conference on Statistics. Mathematics and Related Fields.-Honolulu, 2004.-P. 1-15.
- Grabarnik G. Ya., Katz A. A., Shwartz L. Ergodic type theorems for actions of finitely generated semigroups on von neumann algebras. II//Proceedings of the 4th Annual Hawaii International Conference on Statistics. Mathematics and Related Fields.-Honolulu, 2005.-P. 1-8.
- Grabarnik G. Ya., Katz A. A. On Neveu decomposition and ergodic type theorems for semi-finite von Neumann algebras//Vladikavk. Math. J.-2003.-V. 5, № 2.-P. 5-9.
- Grabarnik G. Ya., Katz A. A. Ergodic type theorems for finite von Neumann algebras//Israel J. of Math.-1995.-V. 90.-P. 403-422.
- Grabarnik G. Ya., Katz A. A. On multiparametric superadditive stochastic ergodic theorem for semi-finite von Neumann algebras, in preparation.
- Grigorchuk R. I. Individual ergodic theorem for actions of free groups//Proceedings of the Tambov workshop in the theory of functions.-1986.-P. 3-15.
- Grigorchuk R. I. Ergodic theorems for actions of free groups and free semigroups//Math. Notes.-1999.-V. 65, № 5.-P. 654-657.-In Russian; translation from Math. Zametki.-1999.-V. 65, № 5.-P. 779-783.-In English.
- Guivarch Y. Generalisation dun theoreme de von Neumann//C. R. Acad. Sci. Paris Ser. A-B.-1969.-V. 268.-P. A1020-A1023.-In French.
- Jajte R. Strong limit theorem in noncommutative probability.-Berlin etc.: Spring, 1985.-162 p. (Lecture Notes in Math.; 1110).
- Hajian A., Kakutani S. Weakly wandering sets and invariant measures//Transactions of the American Mathematical Society.-1964.-V. 110.-P. 131-151.
- Kakutani S. Random ergodic theorems and Markoff processes with a stable distribution//Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950.-P. 247-261.-Berkeley and Los Angeles: University of California Press, 1951.
- Katz A. A. Ergodic type theorems in von Neumann algebras.-Pretoria, University of South Africa, 2001.-P. 84.-(Ph. D. Thesis University of South Africa).
- Kingman J. F. C. Subadditive ergodic theory//Annals of Probability. 1.-1973.-P. 883-909.
- Kovacs I., Szucs J. Ergodic type theorem in von Neumann algebras//Acta Scientiarum Mathematicarum (Szeged).-1966.-V. 27-P. 233-246.
- Krengel U. Ergodic theorems. With a supplement by Antoine Brunel. de Gruyter Studies in Mathematics, 6.-Berlin: Walter de Gruyter & Co, 1985.-357 p.
- Lance E. C. Ergodic theorems for convex sets and operator algebras//Inventiones Mathematicae.-1976.-V. 37.-P. 201-214.
- Lorch E. R. Means of iterated transformations in reflexive vector spaces//Bull. Amer. Math. Soc.-1938.-V. 45.-P. 945-947.
- Nevo A. Harmonic analysis and pointwise ergodic theorems for noncommuting transformations//J. Amer. Math. Soc.-1994.-V. 7, № 4.-P. 875-902.
- Nevo A., Stein E. M. A generalization of Birkhoff's pointwise ergodic theorem//Acta Math.-1994.-V. 173, № 1.-P. 135-154.
- Oseledec V. I. Markov chains, skew products and ergodic theorems for > dynamic systems//Teor. Verojatnost. i Primenen.-1965.-V. 10.-P. 551-557.-In Russian.
- Pisier G., Xu Q. Non-Commutative L_p-Spaces, Handbook of the Geometry of Banach Spaces.-North-Holland, Amsterdam, 2003.-V. 2.-P. 1459-1517.
- Segal I. E. A noncommutative extension of abstract integration//Archiv der Mathematik.-1953.-V. 57.-P. 401-457.
- Sinai Ya. G., Anshelevich V. V. Some problems of noncommutative ergodic theory//Uspekhi Math. Nauk.-1976.-V. 32.-P. 157-174.
- Vershik A. M. Numerical characteristics of groups and relations between them//Zap. Nauchn. Semin. POMI.-1999.-V. 256.-P. 5-18.
- Viennot G. X. Heaps of pieces. I. Basic definitions and combinatorial lemmas. Graph theory and its applications: East and West (Jinan, 1986).-P. 542-570.-New York: Acad. Sci., 1989.-(Ann. New York Acad. Sci.; 576).
- Yeadon F. J. Ergodic theorems for semi-finite von Neumann algebras, I//Journal of the London Mathematical Society.-1977.-V. 16.-P. 326-332.
- Yeadon F. J. Ergodic theorems for semi-finite von Neumann algebras, II//Mathematical Proceedings of the Cambridge Philosophical Society.-1980.-V. 88.-P. 135-147.
