On non-commutative ergodic type theorems for free finitely generated semigroups

G. Ya. Grabarnik   ; A. A. Katz   ; L. A. Shwartz

Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 1, pp. 38-47

Résumé

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.

Bibliographie

[1] Akcoglu M. A., “A pointwise ergodic theorem in $L_{p}$-spaces”, Canad. J. Math., 27:5 (1975), 1075–1082 | MR | Zbl

[2] Russian Math. Surveys, 54:4 (1999), 835–836 | DOI | MR | Zbl

[3] Funct. Anal. Appl., 34:4 (2000), 239–251 | DOI | MR | Zbl

[4] Dixmier J., Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann), Reprint of the second (1969) edition, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Paris, 1996, 367 pp. (In French) | MR

[5] Goldstein M. Sh., “Theorems of almost everywhere convergence in von Neumann algebras”, J. Oper. Theory, 6 (1981), 233–311 (In Russian) | MR

[6] Goldstein M. Sh., Grabarnik G. Ya., “Almost sure convergence theorems in von Neumann algebras”, Israel J. Math., 76:1/2 (1991), 161–182 | DOI | MR | Zbl

[7] Grabarnik G. Ya., Katz A. A., Shwartz L., “Ergodic type theorems for actions of finitely generated semigroups on von neumann algebras, I”, Mathematics and Related Fields, Proceedings of the 3rd Annual Hawaii International Conference on Statistics, Honolulu, 2004, 1–15

[8] Grabarnik G. Ya., Katz A. A., Shwartz L., “Ergodic type theorems for actions of finitely generated semigroups on von neumann algebras, II”, Mathematics and Related Fields, Proceedings of the 4th Annual Hawaii International Conference on Statistics, Honolulu, 2005, 1–8

[9] Grabarnik G. Ya., Katz A. A., “On Neveu decomposition and ergodic type theorems for semi-finite von Neumann algebras”, Vladikavk. Math. J., 5:2 (2003), 5–9 | MR | Zbl

[10] Grabarnik G. Ya., Katz A. A., “Ergodic type theorems for finite von Neumann algebras”, Israel J. of Math., 90 (1995), 403–422 | DOI | MR | Zbl

[11] Grabarnik G. Ya., Katz A. A., On multiparametric superadditive stochastic ergodic theorem for semi-finite von Neumann algebras, in preparation

[12] Grigorchuk R. I., “Individual ergodic theorem for actions of free groups”, Proceedings of the Tambov workshop in the theory of functions, 1986, 3–15

[13] Mat. Zametki, 65:5 (1999), 779–783 (In Russian) | DOI | MR | Zbl

[14] Guivarc'h Y., “Généralisation d'un théorème de von Neumann”, C. R. Acad. Sci. Paris Sér. A-B, 268 (1969), A1020–A1023 (In French) | MR

[15] Jajte R., Strong limit theorem in non-commutative probability, Lecture Notes in Math., 1110, Spring, Berlin etc., 1985, 162 pp. | MR | Zbl

[16] Hajian A., Kakutani S., “Weakly wandering sets and invariant measures”, Transactions of the American Mathematical Society, 110 (1964), 131–151 | MR

[17] Kakutani S., “Random ergodic theorems and Markoff processes with a stable distribution”, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (1950), University of California Press, Berkeley–Los Angeles, 1951, 247–261 | MR

[18] Katz A. A., Ergodic type theorems in von Neumann algebras, Ph. D. Thesis University of South Africa, University of South Africa, Pretoria, 2001, 84 pp.

[19] Kingman J. F. C., “Subadditive ergodic theory”, Annals of Probability, 1 (1973), 883–909 | DOI | MR | Zbl

[20] Kovacs I., Szücs J., “Ergodic type theorem in von Neumann algebras”, Acta Scientiarum Mathematicarum (Szeged), 27 (1966), 233–246 | MR | Zbl

[21] Krengel U., Ergodic theorems, With a supplement by Antoine Brunel, de Gruyter Studies in Mathematics, 6, Walter de Gruyter Co, Berlin, 1985, 357 pp. | MR | Zbl

[22] Lance E. C., “Ergodic theorems for convex sets and operator algebras”, Inventiones Mathematicae, 37 (1976), 201–214 | DOI | MR | Zbl

[23] Lorch E. R., “Means of iterated transformations in reflexive vector spaces”, Bull. Amer. Math. Soc., 45 (1938), 945–947 | DOI | MR

[24] Nevo A., “Harmonic analysis and pointwise ergodic theorems for noncommuting transformations”, J. Amer. Math. Soc., 7:4 (1994), 875–902 | MR | Zbl

[25] Nevo A., Stein E. M., “A generalization of Birkhoff's pointwise ergodic theorem”, Acta Math., 173:1 (1994), 135–154 | DOI | MR | Zbl

[26] Oseledec V. I., “Markov chains, skew products and ergodic theorems for “general” dynamic systems”, Teor. Verojatnost. i Primenen., 10 (1965), 551–557 (In Russian) | MR | Zbl

[27] Pisier G., Xu Q., “Non-Commutative $L_p$-Spaces”, Handbook of the Geometry of Banach Spaces, v. 2, North-Holland, Amsterdam, 2003, 1459–1517 | MR | Zbl

[28] Segal I. E., “A non-commutative extension of abstract integration”, Archiv der Mathematik, 57 (1953), 401–457 | MR | Zbl

[29] Sinai Ya. G., Anshelevich V. V., “Some problems of non-commutative ergodic theory”, Uspekhi Math. Nauk, 31:4(190) (1976), 151–167 | MR | Zbl

[30] Vershik A. M., “Numerical characteristics of groups and relations between them”, Zap. Nauchn. Semin. POMI, 256, 1999, 5–18 | MR | Zbl

[31] Viennot G. X., “Heaps of pieces. I: Basic definitions and combinatorial lemmas”, Graph theory and its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci., 576, Acad. Sci., New York, 1989, 542–570 | MR | Zbl

[32] Yeadon F. J., “Ergodic theorems for semi-finite von Neumann algebras, I”, Journal of the London Mathematical Society, 16 (1977), 326–332 | DOI | MR | Zbl

[33] Yeadon F. J., “Ergodic theorems for semi-finite von Neumann algebras, II”, Mathematical Proceedings of the Cambridge Philosophical Society, 88 (1980), 135–147 | DOI | MR | Zbl