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

Lau, Anthony; Losert, Viktor

doi:10.1090/S0002-9947-99-02242-4

Ergodic sequences in the Fourier-Stieltjes algebra and measure algebra of a locally compact group
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by Anthony To-Ming Lau and Viktor Losert PDF

Trans. Amer. Math. Soc. 351 (1999), 417-428 Request permission

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

Julius Blum and Bennett Eisenberg, Generalized summing sequences and the mean ergodic theorem, Proc. Amer. Math. Soc. 42 (1974), 423–429. MR 330412, DOI 10.1090/S0002-9939-1974-0330412-0
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628, DOI 10.24033/bsmf.1607
E. 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), no. 3, 459–472. MR 722579, DOI 10.1216/RMJ-1981-11-3-459
Anthony To Ming Lau, The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 267 (1981), no. 1, 53–63. MR 621972, DOI 10.1090/S0002-9947-1981-0621972-9
Anthony To Ming Lau and Viktor Losert, The $C^*$-algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993), no. 1, 1–30. MR 1207935, DOI 10.1006/jfan.1993.1024
V. Losert and H. Rindler, Uniform distribution and the mean ergodic theorem, Invent. Math. 50 (1978/79), no. 1, 65–74. MR 516604, DOI 10.1007/BF01406468
Paul Milnes and Alan L. T. Paterson, Ergodic sequences and a subspace of $\textbf {B}(G)$, Rocky Mountain J. Math. 18 (1988), no. 3, 681–694. MR 972658, DOI 10.1216/RMJ-1988-18-3-681
W. Maxones and H. Rindler, Einige Resultate über unitär gleichverteilte Massfolgen, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 2 (1977), 11–13 (German). MR 486304
I. Namioka, A substitute for Lebesgue’s bounded convergence theorem, Proc. Amer. Math. Soc. 12 (1961), 713–716. MR 125427, DOI 10.1090/S0002-9939-1961-0125427-2
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
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
Harald Rindler, Gleichverteilte Folgen in lokalkompakten Gruppen, Monatsh. Math. 82 (1976), no. 3, 207–235. MR 427535, DOI 10.1007/BF01526327
P. F. Renaud, Invariant means on a class of von Neumann algebras, Trans. Amer. Math. Soc. 170 (1972), 285–291. MR 304553, DOI 10.1090/S0002-9947-1972-0304553-0
Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978, DOI 10.1007/978-1-4684-9928-5
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728, DOI 10.1007/978-1-4612-6188-9
K. Gröchenig, V. Losert, and H. Rindler, Uniform distribution in solvable groups, Probability measures on groups, VIII (Oberwolfach, 1985) Lecture Notes in Math., vol. 1210, Springer, Berlin, 1986, pp. 97–107. MR 878998, DOI 10.1007/BFb0077176