Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Pointwise ergodic theorems for bounded Lamperti representations of amenable groups


Author: A. Tempelman
Journal: Proc. Amer. Math. Soc. 143 (2015), 4989-5004
MSC (2010): Primary 22D40; Secondary 37A30
DOI: https://doi.org/10.1090/proc/12616
Published electronically: April 3, 2015
MathSciNet review: 3391055
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Dominated and Pointwise Ergodic Theorems for bounded representations of second countable locally compact amenable groups by Lamperti operators in $ L^p(\Omega ,\mathcal F, m),$ $ p>1$ fixed, are proved; we restrict ourselves to Cesàro averages in this paper. These theorems generalize or are closely related to well-known theorems for powers of power bounded Lamperti operators.


References [Enhancements On Off] (What's this?)

  • [1] M. A. Akcoglu, A pointwise ergodic theorem in $ L_{p}$-spaces, Canad. J. Math. 27 (1975), no. 5, 1075-1082. MR 0396901 (53 #761)
  • [2] Nakhlé Asmar, Earl Berkson, and T. A. Gillespie, Transference of strong type maximal inequalities by separation-preserving representations, Amer. J. Math. 113 (1991), no. 1, 47-74. MR 1087801 (92b:43006), https://doi.org/10.2307/2374821
  • [3] I. Assani, Sur les opérateurs à puissances bornées et le théorème ergodique ponctuel dans $ L^p[0,1]$, $ 1<p<+\infty $, Canad. J. Math. 38 (1986), no. 4, 937-946 (French). MR 854147 (87k:47015), https://doi.org/10.4153/CJM-1986-046-6
  • [4] A. P. Calderon, A general ergodic theorem, Ann. of Math. (2) 58 (1953), 182-191. MR 0055415 (14,1071a)
  • [5] Mischa Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105-167 (1956) (English, with Spanish summary). MR 0084632 (18,893d)
  • [6] Mahlon Day M., Ergodic theorems for Abelian semigroups, Trans. Amer. Math. Soc. 51 (1942), 399-412. MR 0006614 (4,14b)
  • [7] N. Dunford and J. T. Schwartz, Linear operators, Part I (1988), John Wiley & Sons Inc., New York.
  • [8] William R. Emerson, The pointwise ergodic theorem for amenable groups, Amer. J. Math. 96 (1974), 472-487. MR 0354926 (50 #7403)
  • [9] Moshe Feder, On power-bounded operators and the pointwise ergodic property, Proc. Amer. Math. Soc. 83 (1981), no. 2, 349-353. MR 624929 (82j:47013), https://doi.org/10.2307/2043526
  • [10] Frederick P. Greenleaf and William R. Emerson, Group structure and the pointwise ergodic theorem for connected amenable groups, Advances in Math. 14 (1974), 153-172. MR 0384997 (52 #5867)
  • [11] Arshag Hajian and Yuji Ito, Weakly wandering sets and invariant measures for a group of transformations, J. Math. Mech. 18 (1968/1969), 1203-1216. MR 0240283 (39 #1632)
  • [12] Michael Hochman, Averaging sequences and abelian rank in amenable groups, Israel J. Math. 158 (2007), 119-128. MR 2342460 (2008m:43002), https://doi.org/10.1007/s11856-007-0006-x
  • [13] A. Ionescu Tulcea, Ergodic properties of isometries in $ L^{p}$ spaces, $ 1<p<\infty $, Bull. Amer. Math. Soc. 70 (1964), 366-371. MR 0206207 (34 #6026)
  • [14] A. Ionescu Tulcea, On the category of certain classes of transformations in ergodic theory, Trans. Amer. Math. Soc. 114 (1965), 261-279. MR 0179327 (31 #3575)
  • [15] Charn Huen Kan, Ergodic properties of Lamperti operators, Canad. J. Math. 30 (1978), no. 6, 1206-1214. MR 511557 (80g:47037), https://doi.org/10.4153/CJM-1978-100-x
  • [16] Charn-Huen Kan, Ergodic properties of Lamperti operators. II, Canad. J. Math. 35 (1983), no. 4, 577-588. MR 723031 (85g:47012), https://doi.org/10.4153/CJM-1983-033-1
  • [17] Michael Lin and James Olsen, Besicovitch functions and weighted ergodic theorems for LCA group actions, Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 277-289. MR 1412612 (98c:28025)
  • [18] Michael Lin, James Olsen, and Arkady Tempelman, On modulated ergodic theorems for Dunford-Schwartz operators, Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997), 1999, pp. 542-567. MR 1700609 (2000e:47021)
  • [19] Michael Lin and Arkady Tempelman, Averaging sequences and modulated ergodic theorems for weakly almost periodic group representations, J. Anal. Math. 77 (1999), 237-268. MR 1753487 (2001f:22018), https://doi.org/10.1007/BF02791262
  • [20] Michael Lin and Rainer Wittmann, Ergodic sequences of averages of group representations, Ergodic Theory Dynam. Systems 14 (1994), no. 1, 181-196. MR 1268716 (97a:22007), https://doi.org/10.1017/S0143385700007793
  • [21] Elon Lindenstrauss, Pointwise theorems for amenable groups, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 82-90 (electronic). MR 1696824 (2000g:28042), https://doi.org/10.1090/S1079-6762-99-00065-7
  • [22] Elon Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146 (2001), no. 2, 259-295. MR 1865397 (2002h:37005), https://doi.org/10.1007/s002220100162
  • [23] F.J.  Martin-Reyes and A. de la Torre, The dominated ergodic theorem for invertible, positive operators, Semesterbericht Funktionalanalysis Tübingen, Sommersemester 1985, pp. 143-150.
  • [24] F. J. Martín-Reyes and A. De la Torre, The dominated ergodic estimate for mean bounded, invertible, positive operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 69-75. MR 958045 (89i:47015), https://doi.org/10.2307/2047463
  • [25] S. A. McGrath, Some ergodic theorems for commuting $ L_{1}$ contractions, Studia Math. 70 (1981), no. 2, 153-160. MR 642190 (83c:47015)
  • [26] Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264 (86a:43001)
  • [27] H. R. Pitt, Some generalizations of the ergodic theorem, Proc. Cambridge Philos. Soc. 38 (1942), 325-343. MR 0007947 (4,219h)
  • [28] C. Ryll-Nardzewski, Topics in ergodic theory, Probability--Winter School (Proc. Fourth Winter School, Karpacz, 1975), Springer, Berlin, 1975, pp. 131-156. Lecture Notes in Math., Vol. 472. MR 0390177 (52 #11003)
  • [29] A. L. Shulman, Pointwise averaging sequences on groups, Ph.D. Thesis, Vilnius (1988) (in Russian).
  • [30] A. A. Tempelman, Ergodic theorems for general dynamical systems, Dokl. Akad. Nauk SSSR 176 (1967), 790-793 (Russian). MR 0219700 (36 #2779)
  • [31] A. A. Tempelman, Ergodic theorems for general dynamical systems, Trudy Moskov. Mat. Obšč. 26 (1972), 95-132 (Russian). MR 0374388 (51 #10588)
  • [32] A. A. Tempelman, An ergodic theorem for amplitude modulated homogeneous random fields, Litovsk. Mat. Sb. 14 (1974), no. 4, 221-229, 243 (Russian, with Lithuanian and English summaries). MR 0397853 (53 #1709)
  • [33] Arkady Tempelman, Ergodic theorems for group actions, Mathematics and its Applications, vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Informational and thermodynamical aspects; Translated and revised from the 1986 Russian original. MR 1172319 (94f:22007)
  • [34] Romain Tessera, Volume of spheres in doubling metric measured spaces and in groups of polynomial growth, Bull. Soc. Math. France 135 (2007), no. 1, 47-64 (English, with English and French summaries). MR 2430198 (2009h:53086)
  • [35] Norbert Wiener, The ergodic theorem, Duke Math. J. 5 (1939), no. 1, 1-18. MR 1546100, https://doi.org/10.1215/S0012-7094-39-00501-6

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 22D40, 37A30

Retrieve articles in all journals with MSC (2010): 22D40, 37A30


Additional Information

A. Tempelman
Affiliation: Department of Mathematics and Department of Statistics, The Pennsylvania State University, 325 Thomas Building, University Park, Pennsylvania 16802

DOI: https://doi.org/10.1090/proc/12616
Received by editor(s): January 26, 2014
Received by editor(s) in revised form: January 30, 2014, February 3, 2014, and August 20, 2014
Published electronically: April 3, 2015
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society