Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T02:28:11.898Z Has data issue: false hasContentIssue false

Ergodic sequences of averages of group representations

Published online by Cambridge University Press:  19 September 2008

Michael Lin
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev Beer-Sheva, Israel
Rainer Wittmann
Affiliation:
Institut für Mathematische Stochastik, Lotzestrasse 13, Gottingen, Germany

Abstract

Let G be a locally compact σ -compact group with right Haar measure λ. A sequence {μn} of probabilities on G is called ergodic if for every f ∈ L1(G, λ) and t ∈ G we have ‖μn* (f − δt* f) ‖1 → 0. If T (t) is a bounded continuous representation of G by linear operators in a Banach space X, we define the μ,-average of T(t) by .

Type
Research Article
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[A]Azencott, R.. Espaces de Poisson des groupes localement compactes. Springer Lecture Notes in Mathematics 148. Springer, Berlin, 1970.Google Scholar
[AIBi]Alaoglu, L. & Birkhoff, G.. General ergodic theorems. Ann. Math. 41 (1940), 293309.CrossRefGoogle Scholar
[BJR1]Bellow, A., Jones, R. & Rosenblatt, J.. Almost everywhere convergence of weighted averages. Math. Annalen. 293 (1992), 399–26.CrossRefGoogle Scholar
[BJR2]Bellow, A., Jones, R. & Rosenblatt, J.. Almost everywhere convergence of powers. Almost Everywhere Convergence. Academic, New York, 1989. pp. 99120.Google Scholar
[BJR3]Bellow, A., Jones, R. & Rosenblatt, J.. Almost everywhere convergence of convolution powers. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[BlEi]Blum, J. R. & Eisenberg, B.. Generalized summing sequences and the mean ergodic theorem. Proc. Amer. Math. Soc. 42 (1979), 423429.CrossRefGoogle Scholar
[BIRei]Blum, J. R. & Reich, J. I.. Pointwise ergodic theorems in lea groups. Pacific J. Math. 103 (1982), 301306.CrossRefGoogle Scholar
[C]P, A.. Calderon. Ergodic theory and translation-invariant operators. Proc. Nat. Acad. Sci. USA 595 (1968), 349353.Google Scholar
[D1]Derriennic, Y.. Lois ‘Zéros ou deux’ pour les processus de Markov. Applications aux marches aléatoires. Ann. Inst. Poincaré B 12 (1976), 111129.Google Scholar
[D2]Derriennic, Y.. Entropie, théorèmes limites et marches aléatoires. Springer Lecture Notes in Mathematics 1210. Springer, Berlin, 1986. pp. 241284.Google Scholar
[D3]Derriennic, Y.. Entropy and boundary for random walks on locally compact groups. Trans. Tenth Prague Conf. Information Theory etc. Academia, Prague, 1988. pp. 269275.Google Scholar
[DL1]Derriennic, Y. & Lin, M.. Sur la tribu asymptotique de Marches aléatoires sur les groupes. Séminaire de probabilités. (Rennes). 1983.Google Scholar
[DL2]Derriennic, Y. & Lin, M.. Sur le comportement asymptotique des puissances de convolution d'une probabilité. Ann. Inst. Poincaré. B 20 (1984), 127132.Google Scholar
[DL3]Derriennic, Y. & Lin, M.. Convergence of iterates of averages of certain operator representations and of convolution powers. J. Fund. Anal. 85 (1989), 86102.CrossRefGoogle Scholar
[DL4]Derriennic, Y. & Lin, M.. On pointwise convergence in random walks. Almost Everywhere Convergence. Academic, New York, 1989. pp. 189193.Google Scholar
[DL5]Derriennic, Y. & Lin, M.. Uniform ergodic convergence and averaging along Markov chain trajectories. J. Theor. Prob. to appear.Google Scholar
[Da]M, M.. Day. Amenable semi-groups. Illinois J. Math. 1 (1957), 509544.Google Scholar
[DuSc]Dunford, N. & Schwartz, J.. Linear Operators, Part I. Wiley, New York, 1958.Google Scholar
[E]E, W.. Eberlein. Abstract ergodic theorems and weak almost periodic functions. Trans. Amer. Math. Soc. 67 (1949), 217240.Google Scholar
[F1]Foguel, S. R.. On iterates of convolutions. Proc. Amer. Math. Soc. 47 (1975), 368370.CrossRefGoogle Scholar
[F2]Foguel, S. R.. Iterates of a convolution on a non-Abelian group. Ann. Inst. Poincaré B 11 (1975), 199202.Google Scholar
[Fu]Furstenberg, H.. Boundary theory and stochastic processes on homogeneous spaces. Harmonic Analysis on Homogeneous Spaces. Proc. Symp. Pure Math. 26 (1972), 193229.CrossRefGoogle Scholar
[G]Glasner, S.. On Choquet-Deny measures. Ann. Inst. Poincaré B 12 (1976), 110.Google Scholar
[Ga]Garsia, A.M.. Topics in Almost Everywhere Convergence. Markham, Chicago, 1970.Google Scholar
[H]Horowitz, S.. Pointwise convergence of the iterates of a Harris recurrent Markov operator. Israel J. Math. 33 (1979), 177180.CrossRefGoogle Scholar
[HeRo]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis. Vol. I, 2nd edn. Springer, Berlin, 1979.CrossRefGoogle Scholar
[HoM]Hofmann, K. and Mukherjea, A.. Concentration functions and a class of non-compact groups. Math. Ann. 256 (1981), 535548.CrossRefGoogle Scholar
[J]Jones, R.. Ergodic averages on spheres. J. d'Analyse Math. To appear.Google Scholar
[JRT]Jones, R., Rosenblatt, J. & Tempelman, A.. Ergodic theorems for convolutions of a measure on a group. Illinois J. Math. to appear.Google Scholar
[K]Krengel, U.. Ergodic Theorems, de Gruyter, Berlin, 1985.CrossRefGoogle Scholar
[Kav]Kaimanovich, V. A.. & Vershik, A. M.. Random walks on discrete groups: boundary and entropy. Ann. Prob. 11 (1983), 457490.CrossRefGoogle Scholar
[Kan]H, C.. Kan. Ergodic properties for Lamperti operators. Canadian J. Math. 30 (1978), 12061214.Google Scholar
[KeMa]Kerstan, J. & Matthes, K.. Gleichverteilungseigenschaften von Faltungpotenzen auf lokalkompakten Abelschen Gruppen. Wiss. Z. Univ. Jena 14 (1965), 457462.Google Scholar
[L1]Lin, M.. Ergodic properties of an operator obtained from a continuous representation. Ann. Inst. Poincaré B 13 (1977), 321331.Google Scholar
[L2]Lin, M.. Convergence of convolution powers of a probability on a LCA group. Semesterbericht Funktionalanalysis, T¨bingen 3 (1982/3), 110.Google Scholar
[Lo]Lorentz, G. G.. A contribution to the theory of divergent sequences. Acta. Math. 80 (1948), 167190.CrossRefGoogle Scholar
[MPa]Milnes, P. & Paterson, A.. Ergodic sequences and a subspace of B(G). Rocky Mountain J. 18 (1988), 681694.Google Scholar
[NAs]Namioka, I. & Asplund, E.. A geometric proof of Ryll-Nardzewski's fixed point theorem. Bull. Amer. Math. Soc. 73 (1967), 443445.CrossRefGoogle Scholar
[P]P, J.. Pier. Amenable Locally Compact Groups. Wiley, New York, 1984.Google Scholar
[R1]Rosenblatt, J.. Ergodic and mixing random walks on locally compact groups. Math. Ann. 257 (1981), 3142.CrossRefGoogle Scholar
[R2]Rosenblatt, J.. Ergodic group actions. Arch. Math. 47 (1986), 263269.CrossRefGoogle Scholar
[Re]Revuz, D.. Markov Chains. North-Holland, Amsterdam, 1975.Google Scholar
[RyNa]Ryll-Nardzewski, C.. On fixed points of semi-groups of endomorphisms of linear spaces. Proc. Fifth Berkely Symp. Math. Stat. Prob. II (1965/6), 5561.Google Scholar
[S]J, A.. Stam. On shifting iterated convolutions. Compositio Math. 17 (1966), 268280.Google Scholar
[Si]Sine, R.. A note on sequential convergence to invariance. Proc. Amer. Math. Soc. 55 (1976), 313319.Google Scholar
[St]M, E.. Stein. On the maximal ergodic theorem. Proc. Nat. Acad. Sci. USA 47 (1961), 18941897.Google Scholar
[Sta]Starr, N.. Operator limit theorems. Trans. Amer. Math. Soc. 121 (1966), 90115.CrossRefGoogle Scholar
[T1]Tempelman, A.. Ergodic theorems for general dynamical systems, Dokl. Akad. Nauk. SSSR 176 (1967), 790793;Google Scholar
(Sov. Math. Dokl. 8 (1967), 12131216).Google Scholar
[T2]Tempelman, A.. Ergodic theorems for general dynamical systems. Trans. Moscow Math. Soc. (Trudy Moskov) 26 (1972), 94132.Google Scholar
[T3]Tempelman, A.. Ergodic functions and averaging sequences. Dokl. Akad. Nauk SSSR 259 (1981);Google Scholar
(Sov. Math. Dokl. 24 (1981), 7882).Google Scholar
[T4]Tempelman, A.. Ergodic Theorems for Group Actions. Kluwer: Dordrecht, 1991. To appear.Google Scholar
[T5]Tempelman, A.. Ergodic Theorems on Groups. Mokslas, Vilnius, 1985, (in Russian).Google Scholar
[To]de la Torre, A.. A simple proof of the maximal ergodic theorem. Canadian J. Math. 28 (1976), 10731075.CrossRefGoogle Scholar