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Transactions of the American Mathematical Society
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.

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PII: S 0002-9947(1972)0310548-3
Keywords: von Neumann mean, measure algebra, $ {C^ \ast }$-algebra, amenable, trace, dual of a locally compact group
Article copyright: © Copyright 1972 American Mathematical Society