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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Invariant means on almost periodic functions and equicontinuous actions


Author: Anthony To Ming Lau
Journal: Proc. Amer. Math. Soc. 49 (1975), 379-382
MSC: Primary 43A07
DOI: https://doi.org/10.1090/S0002-9939-1975-0367551-5
MathSciNet review: 0367551
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Abstract: Let $ S$ be a topological semigroup such that the almost periodic functions on $ S$ have a left invariant mean (this is the case, for example, when $ S$ has finite intersection property for closed right ideals). Then whenever $ S$ acts equicontinuously on a compact Hausdorff space $ X$, there exists a compact group $ G$ of homeomorphisms acting equicontinuously on a retract $ Y$ of $ X$ such that $ S$ has a common fixed point in $ X$ if and only if $ G$ has a common fixed point in $ Y$. This result generalises some recent work of T. Mitchell. As an application, we show that whenever $ S$ acts equicontinuously on the closed unit interval $ I$, then $ I$ contains a common fixed point for $ S$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0367551-5
Keywords: Invariant means, almost periodic functions, compact groups, topological semigroups, equicontinuous mappings, common fixed points, commuting real functions
Article copyright: © Copyright 1975 American Mathematical Society