On Lau’s conjecture II 󠀼span style=󠀢color:red󠀢󠀾This article has been retracted󠀼/span󠀾

Salame, Khadime

doi:10.1090/proc/14893

On Lau’s conjecture II
This article has been retracted
HTML articles powered by AMS MathViewer

by Khadime Salame

Proc. Amer. Math. Soc. 148 (2020), 1999-2008

DOI: https://doi.org/10.1090/proc/14893

Published electronically: January 29, 2020

PDF | Request permission

Retraction notice: Proc. Amer. Math. Soc. (to appear).

Abstract:

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer.

References

John F. Berglund, Hugo D. Junghenn, and Paul Milnes, Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. MR 999922
Sławomir Borzdyński and Andrzej Wiśnicki, A common fixed point theorem for a commuting family of weak* continuous nonexpansive mappings, Studia Math. 225 (2014), no. 2, 173–181. MR 3303348, DOI 10.4064/sm225-2-4
Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128
Anthony To Ming Lau, Some fixed point theorems and their applications to $W^*$-algebras, Fixed point theory and its applications (Proc. Sem., Dalhousie Univ., Halifax, N.S., 1975) Academic Press, New York-London, 1976, pp. 121–129. MR 445345
Anthony To Ming Lau, Amenability and fixed point property for semigroup of nonexpansive mappings, Fixed point theory and applications (Marseille, 1989) Pitman Res. Notes Math. Ser., vol. 252, Longman Sci. Tech., Harlow, 1991, pp. 303–313 (English, with French summary). MR 1122837
Anthony To Ming Lau and Wataru Takahashi, Invariant means and fixed point properties for non-expansive representations of topological semigroups, Topol. Methods Nonlinear Anal. 5 (1995), no. 1, 39–57. Contributions dedicated to Ky Fan on the occasion of his 80th birthday. MR 1350344, DOI 10.12775/TMNA.1995.003
Anthony T.-M. Lau and Yong Zhang, Fixed point properties for semigroups of nonlinear mappings and amenability, J. Funct. Anal. 263 (2012), no. 10, 2949–2977. MR 2973331, DOI 10.1016/j.jfa.2012.07.013
Anthony To-Ming Lau and Yong Zhang, Fixed point properties of semigroups of non-expansive mappings, J. Funct. Anal. 254 (2008), no. 10, 2534–2554. MR 2406686, DOI 10.1016/j.jfa.2008.02.006
Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630–641. MR 270356
I. Namioka, Følner’s conditions for amenable semi-groups, Math. Scand. 15 (1964), 18–28. MR 180832, DOI 10.7146/math.scand.a-10723
K. Salame, On Lau’s conjecture, Proc. Amer. Math. Soc. (to appear).
Khadime Salame, Amenability and nonlinear flows in dual Banach spaces, J. Fixed Point Theory Appl. 20 (2018), no. 2, Paper No. 88, 17. MR 3800964, DOI 10.1007/s11784-018-0561-2
Khadime Salame, Nonlinear common fixed point properties of semitopological semigroups in uniformly convex spaces, J. Fixed Point Theory Appl. 19 (2017), no. 2, 1041–1057. MR 3659000, DOI 10.1007/s11784-016-0377-x
Khadime Salame, Fixed point properties for semigroups of non-expansive mappings in conjugate Banach spaces, Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 4, 529–544. MR 3579666