Nets of extreme Banach limits
HTML articles powered by AMS MathViewer
- by Rodney Nillsen PDF
- Proc. Amer. Math. Soc. 55 (1976), 347-352 Request permission
Abstract:
Let $N$ be the set of natural numbers and let $\sigma :N \to N$ be an injection having no periodic points. Let ${M_\sigma }$ be the set of $\sigma$-invariant means on ${l_\infty }$. When $f \in {l_\infty }$ let ${\overline d _\sigma }(f) = \sup \lambda (f)$, where the supremum is taken over all $\lambda \in {M_\sigma }$. It is shown that when $f \in {l_\infty }$, there is a sequence $({\lambda _s})_{s = 2}^\infty$ of extreme points of ${M_\sigma }$ which has no extreme weak$^{\ast }$ limit points and such that ${\lambda _s}(f) = {\overline d _\sigma }(f)$ for $s = 2,3, \ldots$. As a consequence, the extreme points of ${M_\sigma }$ are not weak$^{\ast }$ compact.References
- Ching Chou, Minimal sets and ergodic measures for $\beta N\backslash N$, Illinois J. Math. 13 (1969), 777β788. MR 249569
- George Converse, Isaac Namioka, and R. R. Phelps, Extreme invariant positive operators, Trans. Amer. Math. Soc. 137 (1969), 375β385. MR 243370, DOI 10.1090/S0002-9947-1969-0243370-7
- David Dean and Ralph A. Raimi, Permutations with comparable sets of invariant means, Duke Math. J. 27 (1960), 467β479. MR 121663
- J. Peter Duran, Almost convergence, summability and ergodicity, Canadian J. Math. 26 (1974), 372β387. MR 340886, DOI 10.4153/CJM-1974-039-6
- Jason Gait, Transformation groups with no equicontinuous minimal set, Compositio Math. 25 (1972), 87β92. MR 315684
- 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
- Meyer Jerison, The set of all generalized limits of bounded sequences, Canadian J. Math. 9 (1957), 79β89. MR 83697, DOI 10.4153/CJM-1957-012-x
- Rodney Nillsen, Discrete orbits in $\beta N-N$, Colloq. Math. 33 (1975), no.Β 1, 71β81, 160. MR 397303, DOI 10.4064/cm-33-1-71-81
- Robert R. Phelps, Lectures on Choquetβs theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
- Ralph A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963), 81β94. MR 154005
- Ralph A. Raimi, Homeomorphisms and invariant measures for $\beta N-N$, Duke Math. J. 33 (1966), 1β12. MR 198450
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 55 (1976), 347-352
- MSC: Primary 43A07
- DOI: https://doi.org/10.1090/S0002-9939-1976-0407530-3
- MathSciNet review: 0407530