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The existence of left averaging functions that are not right averaging

Author: Tianxuan Miao
Journal: Proc. Amer. Math. Soc. 115 (1992), 121-123
MSC: Primary 43A07
MathSciNet review: 1079704
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Abstract: Let $ G$ be a locally compact group. We show that $ G$ is amenable as a discrete group if and only if $ \sum\nolimits_{i = 1}^n {{\lambda _{i{x_i}}}} f \in {\mathcal{A}_0}$ for any $ {f_0} \in {\mathcal{A}_0},{x_i} \in G$, and $ {\lambda _i} > 0(i = 1,2, \ldots ,n)$ with $ \sum\nolimits_{i = 1}^n {{\lambda _i} = 1} $, where $ {\mathcal{A}_0}$ is the set of functions that left average to 0. We also confirm a conjecture of Rosenblatt and Yang that there is a left averaging function that is not right averaging if $ G$ is not amenable.

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

  • [1] F. P. Greenleaf, Invariant means on topological groups, Van Nostrand, New York, 1969. MR 0251549 (40:4776)
  • [2] T. Miao, Amenability of locally compact groups and subspaces of $ {L^\infty }(G)$, Proc. Amer. Math Soc. (to appear). MR 1045143 (92d:43003)
  • [3] J. M. Rosenblatt and Z. Yang, Functions with a unique invariant mean value, Illinois J. Math. 34 (1990), 744-764. MR 1062773 (91f:43003)

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Keywords: Locally compact group, amenable group, invariant means, the left averaging functions
Article copyright: © Copyright 1992 American Mathematical Society

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