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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The exact cardinality of the set of topological left invariant means on an amenable locally compact group


Authors: Anthony To Ming Lau and Alan L. T. Paterson
Journal: Proc. Amer. Math. Soc. 98 (1986), 75-80
MSC: Primary 43A07; Secondary 43A15
MathSciNet review: 848879
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Abstract: The purpose of this note is to prove that if $ G$ is an amenable locally compact noncompact group, then the set of topological left invariant means on $ {L_\infty }(G)$ has cardinality $ {2^{{2^d}}}$, where $ d$ is the smallest cardinality of the covering of $ G$ by compact sets. We also prove that in this case the spectrum of the bounded left uniformly continuous complex-valued functions contains exactly $ {2^{{2^d}}}$ minimal closed invariant subsets (or left ideals)


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0848879-6
PII: S 0002-9939(1986)0848879-6
Keywords: Amenable locally compact groups, invariant means, left thick subsets, uniformly continuous functions, minimal invariant sets
Article copyright: © Copyright 1986 American Mathematical Society