The existence of left averaging functions that are not right averaging
HTML articles powered by AMS MathViewer
- by Tianxuan Miao PDF
- Proc. Amer. Math. Soc. 115 (1992), 121-123 Request permission
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
- Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
- Tianxuan Miao, Amenability of locally compact groups and subspaces of $L^\infty (G)$, Proc. Amer. Math. Soc. 111 (1991), no. 4, 1075–1084. MR 1045143, DOI 10.1090/S0002-9939-1991-1045143-1
- Joseph Rosenblatt and Zhuocheng Yang, Functions with a unique mean value, Illinois J. Math. 34 (1990), no. 4, 744–764. MR 1062773
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 121-123
- MSC: Primary 43A07
- DOI: https://doi.org/10.1090/S0002-9939-1992-1079704-1
- MathSciNet review: 1079704