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INVARIANT MEANS AND ACTIONS OF SEMITOPOLOGICAL SEMIGROUPS ON COMPLETELY REGULAR SPACES AND APPLICATIONS

Part of: Abstract harmonic analysis Nonlinear operators and their properties

Published online by Cambridge University Press:  10 June 2020

KHADIME SALAME Open the ORCID record for KHADIME SALAME [Opens in a new window]
KHADIME SALAME*
Affiliation:
Diourbel, Senegal email [email protected]
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Abstract

In this paper, we extend the study of fixed point properties of semitopological semigroups of continuous mappings in locally convex spaces to the setting of completely regular topological spaces. As applications, we establish a general fixed point theorem, a convergence theorem and an application to amenable locally compact groups.

Keywords

amenabilitysemigroupscompletely regular spacesfixed point propertiesHaar measure

MSC classification

Primary: 43A07: Means on groups, semigroups, etc.; amenable groups
Secondary: 43A05: Measures on groups and semigroups, etc. 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 162 - 173
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Berglund, J. F., Junghenn, H. D. and Milnes, P., Analysis on Semigroups (John Wiley, New York, 1989).Google Scholar
Burckel, R. B., Weakly Almost Periodic Functions on Semigroups (Gordon and Breach, New York, 1970).Google Scholar
Day, M. M., ‘Amenable semigroups’, Illinois J. Math. 1 (1957), 509–544.CrossRefGoogle Scholar
De Chiffre, M. D., The Haar Measure, BSc Thesis, University of Copenhagen, 2011.Google Scholar
Granirer, E. and Lau, A. T., ‘Invariant means on locally compact groups’, Illinois J. Math. 15 (1971), 249–257.10.1215/ijm/1256052712CrossRefGoogle Scholar
Izzo, A. J., ‘A functional analysis proof of the existence of Haar measure on locally compact abelian groups’, Proc. Amer. Math. Soc. 115 (1992), 581–583.10.1090/S0002-9939-1992-1097346-9CrossRefGoogle Scholar
Izzo, A. J., ‘A simple proof of the existence of Haar measure on amenable groups’, Math. Scand. 120 (2017), 317–319.CrossRefGoogle Scholar
Lau, A. T.-M., ‘Invariant means on almost periodic functions and fixed point properties’, Rocky Mountain J. Math. 3 (1973), 69–76.10.1216/RMJ-1973-3-1-69CrossRefGoogle Scholar
Lau, A. T.-M. and Zhang, Y., ‘Fixed point properties of semigroups of non-expansive mappings’, J. Funct. Anal. 254 (2008), 2534–2554.10.1016/j.jfa.2008.02.006CrossRefGoogle Scholar
Mitchell, T., ‘Topological semigroups and fixed points’, Illinois J. Math. 14 (1970), 630–641.10.1215/ijm/1256052955CrossRefGoogle Scholar
Rudin, W., Functional Analysis, 2nd edn (McGraw-Hill, New York, 1991).Google Scholar
Runde, V., Lectures on Amenability (Springer, Berlin, 2002).CrossRefGoogle Scholar
Salame, K., Amenability and Fixed Point Properties of Semi-Topological Semigroups of Non-Expansive Mappings in Banach Spaces, PhD Thesis, University of Alberta, 2016.10.36045/bbms/1480993585CrossRefGoogle Scholar
Salame, K., ‘A characterization of 𝜎-extremely amenable semitopological semigroups’, J. Nonlinear Convex Anal. 19 (2018), 1443–1458.Google Scholar
Wilson, B., ‘A fixed point theorem and the existence of a Haar measure for hypergroups satisfying conditions related to amenability’, Canad. Math. Bull. 58 (2015), 415–422.CrossRefGoogle Scholar