Averages of unitary representations and weak mixing of random walks

Michael Lin; Rainer Wittmann

Studia Mathematica (1995)

  • Volume: 114, Issue: 2, page 127-145
  • ISSN: 0039-3223

Abstract

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Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; U n converges weakly for every continuous unitary representation of G; U is weakly mixing for any ergodic group action in a probability space. (ii) If μ is ergodic on G metrizable, and U n converges strongly for every unitary representation, then the random walk is weakly mixing: n - 1 ∑ k = 1 n | ⟨ μ k * f , g ⟩ | → 0 for g ∈ L ∞ ( G ) and f ∈ L 1 ( G ) with ʃ fdλ = 0. (iii) Let G be metrizable, and assume that it is nilpotent, or that it has equivalent left and right uniform structures. Then μ is ergodic and strictly aperiodic if and only if the random walk is weakly mixing. (iv) Weak mixing is characterized by the asymptotic behaviour of μ n on U C B l ( G )

How to cite

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Lin, Michael, and Wittmann, Rainer. "Averages of unitary representations and weak mixing of random walks." Studia Mathematica 114.2 (1995): 127-145. <http://eudml.org/doc/216184>.

@article{Lin1995,
abstract = {Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; $U^n$ converges weakly for every continuous unitary representation of G; U is weakly mixing for any ergodic group action in a probability space. (ii) If μ is ergodic on G metrizable, and $U^n$ converges strongly for every unitary representation, then the random walk is weakly mixing: $n^\{-1\} ∑_\{k=1\}^n |⟨μ^\{k\}*f,g⟩| → 0$ for $g ∈ L_∞(G)$ and $f ∈ L_\{1\}(G)$ with ʃ fdλ = 0. (iii) Let G be metrizable, and assume that it is nilpotent, or that it has equivalent left and right uniform structures. Then μ is ergodic and strictly aperiodic if and only if the random walk is weakly mixing. (iv) Weak mixing is characterized by the asymptotic behaviour of $μ^n$ on $UCB_\{l\}(G)$},
author = {Lin, Michael, Wittmann, Rainer},
journal = {Studia Mathematica},
keywords = {locally compact -compact group; almost periodicity; ergodicity; unitary representations; convolution operators; metrizable groups; nilpotent groups},
language = {eng},
number = {2},
pages = {127-145},
title = {Averages of unitary representations and weak mixing of random walks},
url = {http://eudml.org/doc/216184},
volume = {114},
year = {1995},
}

TY - JOUR
AU - Lin, Michael
AU - Wittmann, Rainer
TI - Averages of unitary representations and weak mixing of random walks
JO - Studia Mathematica
PY - 1995
VL - 114
IS - 2
SP - 127
EP - 145
AB - Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; $U^n$ converges weakly for every continuous unitary representation of G; U is weakly mixing for any ergodic group action in a probability space. (ii) If μ is ergodic on G metrizable, and $U^n$ converges strongly for every unitary representation, then the random walk is weakly mixing: $n^{-1} ∑_{k=1}^n |⟨μ^{k}*f,g⟩| → 0$ for $g ∈ L_∞(G)$ and $f ∈ L_{1}(G)$ with ʃ fdλ = 0. (iii) Let G be metrizable, and assume that it is nilpotent, or that it has equivalent left and right uniform structures. Then μ is ergodic and strictly aperiodic if and only if the random walk is weakly mixing. (iv) Weak mixing is characterized by the asymptotic behaviour of $μ^n$ on $UCB_{l}(G)$
LA - eng
KW - locally compact -compact group; almost periodicity; ergodicity; unitary representations; convolution operators; metrizable groups; nilpotent groups
UR - http://eudml.org/doc/216184
ER -

References

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  1. [AaLWe] J. Aaronson, M. Lin and B. Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products, Israel J. Math. 33 (1979), 198-224. Zbl0438.28018
  2. [A] R. Azencott, Espaces de Poisson des groupes localement compacts, Lecture Notes in Math. 148, Springer, 1970. Zbl0239.60008
  3. [D] Y. Derriennic, Lois "Zéros ou deux" pour les processus de Markov. Applications aux marches aléatoires, Ann. Inst. H. Poincaré B 12 (1976), 111-129. Zbl0353.60075
  4. [DGu] Y. Derriennic et Y. Guivarc'h, Théorème de renouvellement pour les groupes non-moyennables, C. R. Acad. Sci. Paris Sér. A 277 (1973), 613-615. Zbl0272.60005
  5. [DL_1] Y. Derriennic et M. Lin, Sur la tribu asymptotique de marches aléatoires sur les groupes, Séminaire de Probabilités, Rennes, 1983. 
  6. [DL_2] Y. Derriennic et M. Lin, Convergence of iterates of averages of certain operator representations and of convolution powers, J. Funct. Anal. 85 (1989), 86-102. Zbl0712.22008
  7. [DL_3] Y. Derriennic et M. Lin, On pointwise convergence in random walks, in: Almost Everywhere Convergence, Academic Press, 1989, 189-193. 
  8. [Di] J. Dixmier, Les C*-algèbres et leurs représentations, 2 e éd., Gauthier-Villars, Paris, 1969; English transl.: C*-algebras, North-Holland, 1977. 
  9. [Do] L. Dor, On sequences spanning a complex l 1 space, Proc. Amer. Math. Soc. 47 (1975), 515-516. Zbl0296.46014
  10. [F] S. R. Foguel, Ergodic Theory of Markov Processes, van Nostrand, 1969. 
  11. [G] S. Glasner, On Choquet-Deny measures, Ann. Inst. H. Poincaré B 12 (1976), 1-10. Zbl0349.60006
  12. [HRo] E. Hewitt and K. Ross, Abstract Harmonic Analysis, Vol. I, 2nd ed., Springer, Berlin, 1979. 
  13. [HoM] K. Hofmann and A. Mukherjea, Concentration functions and a class of non-compact groups, Math. Ann. 256 (1981), 535-548. Zbl0471.60015
  14. [JL_1] L. Jones and M. Lin, Ergodic theorems of weak mixing type, Proc. Amer. Math. Soc. 57 (1976), 50-52. Zbl0338.47004
  15. [JL_2] L. Jones and M. Lin, Unimodular eigenvalues and weak mixing, J. Funct. Anal. 35 (1980), 42-48. Zbl0441.47009
  16. [JRT] R. Jones, J. Rosenblatt and A. Tempelman, Ergodic theorems for convolutions of a measure on a group, Illinois J. Math. 38 (1994), 521-553. Zbl0831.28008
  17. [KaV] V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), 457-490. Zbl0641.60009
  18. [KeMa] J. Kerstan und K. Matthes, Gleichverteilungseigenschaften von Faltungpotenzen auf lokalkompakten Abelschen Gruppen, Wiss. Z. Univ. Jena 14 (1965), 457-462. 
  19. [K] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. 
  20. [L] M. Lin, Ergodic properties of an operator obtained from a continuous representation, Ann. Inst. H. Poincaré B 13 (1977), 321-331. Zbl0383.60071
  21. [LW_1] M. Lin and R. Wittmann, Ergodic sequences of averages of group representations, Ergodic Theory Dynamical Systems 14 (1994), 181-196. Zbl0814.60004
  22. [LW_2] M. Lin and R. Wittmann, Convergence of representation averages and of convolution powers, Israel J. Math. 88 (1994), 125-157. Zbl0815.60007
  23. [Ly] Yu. Lyubich, Introduction to the Theory of Banach Representations of Groups, Birkhäuser, Basel, 1988. 
  24. [P] J. P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984. Zbl0597.43001
  25. [R] J. Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann. 257 (1981), 31-42. Zbl0451.60011
  26. [Ros] H. P. Rosenthal, A characterization of Banach spaces containing l 1 , Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413. Zbl0297.46013
  27. [S] A. J. Stam, On shifting iterated convolutions, Compositio Math. 17 (1966), 268-280. Zbl0214.43704
  28. [T] A. Tempelman, Ergodic Theorems for Group Actions, Kluwer, Dordrecht, 1992. 
  29. [Wi] G. Willis, The structure of totally disconnected locally compact groups, Math. Ann. 300 (1994), 341-363. Zbl0811.22004
  30. [Z] R. Zimmer, Ergodic Theory and Semi-simple Groups, Birkhäuser, Basel, 1984. Zbl0571.58015

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