On ergodic theorems for free group actions on noncommutative spaces. (English) Zbl 1106.46047
Summary: We extend in a noncommutative setting the individual ergodic theorem of A. Nevo and E. M.Stein [Acta Math.173, No. 1, 135–154 (1994; Zbl 0837.22003)] concerning measure preserving actions of free groups and averages on spheres \(S_{2n}\) of even radius. Here we study state preserving actions of free groups on a von Neumann algebra \(A\) and the behaviour of \((S_{2n}(x))\) for \(x\) in noncommutative spaces \(L^{p}(A)\). For the Cesàro means \(\frac{1}{n}\sum_{k=0}^{n-1}s_k\) and \(p = +\infty\), this problem was solved by T. E.Walker [J. Funct.Anal.150, No. 1, 27–47 (1997; Zbl 0884.22002)]. Our approach is based on ideas of A. I.Bufetov [Funct.Anal.Appl.34, No. 4, 239–251 (2000); translation from Функц.Анал.Прилож. 34, No. 4, 1–17 (2000; Zbl 0983.22006)]. We prove a noncommutative version of Rota’s “Alternierende Verfahren” theorem [cf.G.–C.Rota, Bull.Am.Math.Soc.68, 95–102 (1962; Zbl 0116.10403)]. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.
MSC:
| 46L53 | Noncommutative probability and statistics |
| 46L55 | Noncommutative dynamical systems |
| 47A35 | Ergodic theory of linear operators |
| 28D99 | Measure-theoretic ergodic theory |
References:
| [1] | Accardi, J. Funct. Anal., 45, 245 (1982) · Zbl 0483.46043 · doi:10.1016/0022-1236(82)90022-2 |
| [2] | Arnol’d, Dokl. Akad. Nauk. SSSR, 148, 9 (1963) · Zbl 0237.34008 |
| [3] | Bhat, Ann. Inst. Henri Poincaré, 31, 601 (1995) · Zbl 0832.46060 |
| [4] | Blanchard, Pacific J. Math., 199, 1 (2001) · Zbl 1054.46512 · doi:10.2140/pjm.2001.199.1 |
| [5] | Bratteli, O., Robinson, D.W.: Operator algebras and quantum stastistical mechanics I. Texts and Monographs in Physics, Springer-Verlag, 1979 · Zbl 0421.46048 |
| [6] | Bufetov, Funct. Anal. Appl., 34, 239 (2000) · Zbl 0983.22006 · doi:10.1023/A:1004116205980 |
| [7] | Bufetov, Annals of Math., 155, 929 (2002) · Zbl 1028.37001 |
| [8] | Chilin, V.I., Litvinov, S., Skalski, A.: A few new results in noncommutative ergodic theory. To appear in J. Operator Theory · Zbl 1119.46314 |
| [9] | Connes, Commun. Math. Phys., 112, 691 (1987) · Zbl 0637.46073 · doi:10.1007/BF01225381 |
| [10] | Conze, Invent. Math., 46, 1 (1978) · Zbl 0374.46058 · doi:10.1007/BF01390100 |
| [11] | Dang-Ngoc, Israel J. Math., 34, 273 (1979) · Zbl 0446.60028 |
| [12] | Defant, A., Junge, M.: Classical summation methods in noncommutative probability. In preparation · Zbl 0764.46010 |
| [13] | Goldstein, Math. Zeit., 219, 591 (1995) · Zbl 0828.47036 |
| [14] | Grigorchuk, R.I.: An individual ergodic theorems for actions of free groups. Proc. XII Workshop in the theory of operators in functional spaces. Tambov, 1987 |
| [15] | Guivarc’h, C. R. Acad. Sci. Paris, Ser. A, 268, 1020 (1969) · Zbl 0176.11703 |
| [16] | Jajte, R.: Strong limit theorems in non-commutative probability. Lecture Notes in Math. 1110, Springer-Verlag, 1985 · Zbl 0554.46033 |
| [17] | Jajte, R.: Strong limit theorems in non-commutative L_2-spaces. Lecture Notes in Math. 1477, Springer-Verlag, 1991 · Zbl 0743.46069 |
| [18] | Junge, J. Reine Angew. Math., 549, 149 (2002) · Zbl 1004.46043 |
| [19] | Junge, C. R. Acad. Sci. Paris, Ser. I, 334, 773 (2002) · Zbl 1030.46082 |
| [20] | Junge, M., Xu, Q.: Noncommutative maximal ergodic theorems, Preprint 2004 · Zbl 1116.46053 |
| [21] | Junge, M., Xu, Q.: Non-commutative Burkholder/Rosenthal inequalities. Preprint · Zbl 1041.46050 |
| [22] | Junge, M., Xu, Q.: Haagerup’s reduction on noncommutative L_p-spaces and applications. In preparation |
| [23] | Haagerup, U.: L^p-spaces associated with an arbitrary von Neumann algebra. In: “Algèbres d”opérateurs et leurs applications en physique mathématique”. 175-184, Colloque Internat. CNRS, Marseille 1977 |
| [24] | Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras, Volume II. Graduate Studies in Math. 16, AMS, 1997 · Zbl 0888.46039 |
| [25] | Kaimanovich, V.: Measure-theoretic boundaries of Markov chains, 0 - 2 laws and entropy. In: Picardello, M.A., (ed) Proceedings of the Conference on Harmonic Analysis and Discrete Potential Theory. (Frascati 1991), Plenum 1992, pp 145-180 |
| [26] | Kosaki, J. Funct. Anal., 56, 29 (1984) · Zbl 0604.46063 · doi:10.1016/0022-1236(84)90025-9 |
| [27] | Kümmerer, Invent. Math., 46, 139 (1978) · Zbl 0379.46060 · doi:10.1007/BF01393251 |
| [28] | Lance, Invent. Math., 37, 201 (1975) · Zbl 0338.46054 · doi:10.1007/BF01390319 |
| [29] | Nelson, J. Funct. Anal., 15, 29 (1974) · Zbl 0292.46030 · doi:10.1016/0022-1236(74)90014-7 |
| [30] | Neveu, J.: Bases mathématiques du calcul des probabilités. Masson, 1970 · Zbl 0203.49901 |
| [31] | Nevo, A., Harmonic analysis and pointwise ergodic theorems for noncommuting transformations, J. Amer. Math. Soc., 7, 875-902 (1994) · Zbl 0812.22005 |
| [32] | Nevo, Acta Math., 173, 135 (1994) · Zbl 0837.22003 · doi:10.1007/BF02392571 |
| [33] | Paschke, Trans. Amer. Math. Soc., 182, 443 (1973) · Zbl 0239.46062 · doi:10.2307/1996542 |
| [34] | Pisier, G.: Non-commutative vector valued L_p-spaces and completely p-summing maps. Astérisque, 247, 1998 · Zbl 0937.46056 |
| [35] | Revuz, D.: Markov chains. North-Holland, 1984 · Zbl 0539.60073 |
| [36] | Rota, Bull. A. M. S., 68, 95 (1962) · Zbl 0116.10403 · doi:10.1090/S0002-9904-1962-10737-X |
| [37] | Sauvageot, J-L.: A note on almost uniform convergence in von Neumann algebras. In: “Quantum probability and related topics”. Word Sc. Publ., 1991, pp 385-390 · Zbl 0928.46052 |
| [38] | Sauvageot, Commun. Math. Phys., 106, 91 (1986) · Zbl 0606.60079 · doi:10.1007/BF01210927 |
| [39] | Segal, Annals of Math., 57, 401 (1953) · Zbl 0051.34201 · doi:10.2307/1969729 |
| [40] | Stein, Proc. Nat. Acad. Sci. U.S.A., 47, 1894 (1961) · Zbl 0182.47102 |
| [41] | Takesaki, J. Funct. Anal., 9, 306 (1972) · Zbl 0245.46089 · doi:10.1016/0022-1236(72)90004-3 |
| [42] | Takesaki, M.: Theory of operator algebras II. Encyclopaedia of mathematical sciences, 125, Operator algebras and non-commutative geometry. 6, Springer-Verlag, Berlin, 2003 · Zbl 1059.46031 |
| [43] | Terp, J. Operator Theory, 8, 327 (1982) · Zbl 0532.46035 |
| [44] | Voiculescu, D.: Symmetries of some reduced free product C^★-algebras. In: “Operator algebras and their connections with topology and ergodic theory”. 556-588, Lecture Notes in Math. 1132, Springer-Verlag, 1985 · Zbl 0618.46048 |
| [45] | Walker, J. Funct. Anal., 150, 27 (1997) · Zbl 0884.22002 · doi:10.1006/jfan.1997.3105 |
| [46] | Yeadon, J. London Math. Soc., 16, 326 (1977) · Zbl 0369.46061 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
