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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 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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1079704-1
PII: S 0002-9939(1992)1079704-1
Keywords: Locally compact group, amenable group, invariant means, the left averaging functions
Article copyright: © Copyright 1992 American Mathematical Society