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Ergodic sequences in the Fourier-Stieltjes
algebra and measure algebra of
a locally compact group

Authors: Anthony To-Ming Lau and Viktor Losert
Journal: Trans. Amer. Math. Soc. 351 (1999), 417-428
MSC (1991): Primary 43A05, 43A35
MathSciNet review: 1487622
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Abstract: Let $G$ be a locally compact group. Blum and Eisenberg proved that if $G$ is abelian, then a sequence of probability measures on $G$ is strongly ergodic if and only if the sequence converges weakly to the Haar measure on the Bohr compactification of $G.$ In this paper, we shall prove an extension of Blum and Eisenberg's Theorem for ergodic sequences in the Fourier-Stieltjes algebra of $G.$ We shall also give an improvement to Milnes and Paterson's more recent generalization of Blum and Eisenberg's result to general locally compact groups, and we answer a question of theirs on the existence of strongly (or weakly) ergodic sequences of measures on $G.$

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

  • [1] J. Blum and B. Eisenberg, Generalized summing sequences and the mean ergodic theorem, Proc. Amer. Math. Soc. 42 (1974), 423-429. MR 48:8749
  • [2] J. Dixmier, $C^{*}\text{-Algebras}$, North-Holland, Amsterdam - New York - Oxford, 1977. MR 56:16388
  • [3] P. Eymard, L'algèbre de Fourier d'une groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 37:4208
  • [4] E. Granirer and M. Leinert, On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra $B(G)$ and of the measure algebra $M(G)$, Rocky Mountain J. Math. 11 (1981), 459-472. MR 85f:43009
  • [5] A.T. Lau, The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 267 (1981), 53-63. MR 83e:43009
  • [6] A.T. Lau and V. Losert, The $C^{*}\text{-algebra}$ generated by operators with compact support on a locally compact group, Journal of Functional Analysis 112 (1993), 1-30. MR 94d:22005
  • [7] V. Losert and H. Rindler, Uniform distribution and the mean ergodic theorem, Inventiones Math. 50 (1978), 65-74. MR 80f:22001
  • [8] P. Milnes and A. Paterson, Ergodic sequences and a subspace of $B(G)$, Rocky Mountain Journal of Mathematics 18 (1988), 681-694. MR 90a:43002
  • [9] W. Maxones and H. Rindler, Einige Resultate ueber unitär gleichverteilte Massfolgen, Anz. Österreich Akad. Wiss., Math.-Natur. Kl. (1977/2), 11-13. MR 58:6063
  • [10] I. Namioka, A substitute for Lebesgue's bounded convergence theorem, Proc. Amer. Math. Soc. 12 (1961), 713-716. MR 23:A2729
  • [11] A.T. Paterson, Amenability, Mathematical Surveys Monographs, Vol. 29, Amer. Math. Soc. Providence, R.I., 1988. MR 90e:43001
  • [12] J.P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984. MR 86a:43001
  • [13] H. Rindler, Gleichverteilte Folgen in lokalkompakten Gruppen, Monatsh. Math. 82 (1976), 207-235. MR 55:567
  • [14] P.F. Renaud, Invariant means on a class of von Neumann algebras, Trans. Amer. Math. Soc. 170 (1972), 285-291. MR 46:3688
  • [15] H.H. Schaefer, Topological Vector Spaces, Springer-Verlag, New York-Heidelberg-Berlin, 1971. MR 49:7722
  • [16] M. Takesaki, Theory of Operator Algebras I, Springer, New York-Heidelberg-Berlin, 1979. MR 81e:46038
  • [17] K. Gröchenig, V. Losert, H. Rindler, Uniform distribution in solvable groups, Probability Measures on Groups VIII, Proceedings, Oberwolfach, Lecture Notes in Mathematics 1210, Springer, Berlin-Heidelberg-New York, 1986, pp. 97-107. MR 88d:22010

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Additional Information

Anthony To-Ming Lau
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Viktor Losert
Affiliation: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

Keywords: Ergodic sequences, Fourier-Stieltjes algebra, measure algebra, amenable groups
Received by editor(s): February 3, 1997
Additional Notes: This research is supported by NSERC Grant A7679
Article copyright: © Copyright 1999 American Mathematical Society

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