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Proceedings of the American Mathematical Society

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Nets of extreme Banach limits


Author: Rodney Nillsen
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
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0407530-3
Keywords: Motions, means, invariant means, Banach limits, extreme points, ergodic measure, Stone-Čech compactification
Article copyright: © Copyright 1976 American Mathematical Society

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