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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ C\sp{\ast} $-algebras generated by Fourier-Stieltjes transforms

Authors: Charles F. Dunkl and Donald E. Ramirez
Journal: Trans. Amer. Math. Soc. 164 (1972), 435-441
MSC: Primary 43A30; Secondary 22D25, 46L05
MathSciNet review: 0310548
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Abstract: For G a locally compact group and $ \hat G$ its dual, let $ {\mathcal{M}_d}(\hat G)$ be the $ {C^ \ast }$-algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on $ {\mathcal{M}_d}(\hat G)$ is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in $ {\mathcal{M}_d}(\hat G)$ are the discrete measures, and also the sup and lim sup norms are identical on $ {\mathcal{M}_d}(\hat G)$. These results are extensions of classical theorems on almost periodic functions on locally compact abelian groups.

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

  • [1] J. Dixmier, Les $ {C^ \ast }$-algèbres et leurs representations, Cahiers Scientifiques, fasc. 29, Gauthier-Villars, Paris, 1964. MR 30 #1404. MR 0171173 (30:1404)
  • [2] C. Dunkl and D. Ramirez, Helson sets in compact and locally compact groups, Michigan Math. J. 19 (1972). MR 0324315 (48:2667)
  • [3] -, $ {C^ \ast }$-algebras generated by measures, Bull. Amer. Math. Soc. 77 (1971), 411-412. MR 0273418 (42:8297)
  • [4] F. Greenleaf, Invariant means of topological groups and their applications, Van Nostrand Math. Studies, no. 16, Van Nostrand, New York, 1969. MR 40 #4776. MR 0251549 (40:4776)
  • [5] M. Naimark, Normed rings, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1959. MR 19, 870; MR 22 #1824.
  • [6] H. Reiter, Classical harmonic analysis and locally compact groups, Oxford Univ. Press, Oxford, 1967. MR 0306811 (46:5933)

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Keywords: von Neumann mean, measure algebra, $ {C^ \ast }$-algebra, amenable, trace, dual of a locally compact group
Article copyright: © Copyright 1972 American Mathematical Society

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