Functions with a unique mean value and amenability
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- by Yuji Takahashi PDF
- Proc. Amer. Math. Soc. 121 (1994), 775-777 Request permission
Abstract:
It is shown that there exist many amenable locally compact groups for which the sets of functions with unique left invariant mean values are not closed under addition. This resolves negatively a problem raised by T. Miao.References
- V. G. Drinfelโฒd, Finitely-additive measures on $S^{2}$ and $S^{3}$, invariant with respect to rotations, Funktsional. Anal. i Prilozhen. 18 (1984), no.ย 3, 77 (Russian). MR 757256
- G. A. Margulis, Some remarks on invariant means, Monatsh. Math. 90 (1980), no.ย 3, 233โ235. MR 596890, DOI 10.1007/BF01295368
- Tianxuan Miao, Amenability of locally compact groups and subspaces of $L^\infty (G)$, Proc. Amer. Math. Soc. 111 (1991), no.ย 4, 1075โ1084. MR 1045143, DOI 10.1090/S0002-9939-1991-1045143-1
- Joseph Rosenblatt, Uniqueness of invariant means for measure-preserving transformations, Trans. Amer. Math. Soc. 265 (1981), no.ย 2, 623โ636. MR 610970, DOI 10.1090/S0002-9947-1981-0610970-7
- Joseph Rosenblatt, Translation-invariant linear forms on $L_p(G)$, Proc. Amer. Math. Soc. 94 (1985), no.ย 2, 226โ228. MR 784168, DOI 10.1090/S0002-9939-1985-0784168-5
- Joseph Rosenblatt and Zhuocheng Yang, Functions with a unique mean value, Illinois J. Math. 34 (1990), no.ย 4, 744โ764. MR 1062773
- Dennis Sullivan, For $n>3$ there is only one finitely additive rotationally invariant measure on the $n$-sphere defined on all Lebesgue measurable subsets, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no.ย 1, 121โ123. MR 590825, DOI 10.1090/S0273-0979-1981-14880-1
- G. A. Willis, Continuity of translation invariant linear functionals on $C_0(G)$ for certain locally compact groups $G$, Monatsh. Math. 105 (1988), no.ย 2, 161โ164. MR 930434, DOI 10.1007/BF01501168
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 775-777
- MSC: Primary 43A07; Secondary 22C05, 43A15
- DOI: https://doi.org/10.1090/S0002-9939-1994-1189549-1
- MathSciNet review: 1189549