On Lau’s conjecture 󠀼span style=󠀢color:red󠀢󠀾This article has been retracted󠀼/span󠀾

Salame, Khadime

doi:10.1090/proc/14709

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This article has been retracted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

by Khadime Salame

Proc. Amer. Math. Soc. 148 (2020), 343-350

DOI: https://doi.org/10.1090/proc/14709

Published electronically: July 30, 2019

PDF | Request permission

Corrigendum: Proc. Amer. Math. Soc. 152 (2024), 899-900.
Retraction notice: Proc. Amer. Math. Soc. (to appear).

Abstract:

In 1976 during a conference in Halifax, A. Lau raised the question on whether left amenability property for a semitopological semigroup is sufficient to ensure the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex set in a dual Banach space. In this paper we establish a positive answer for discrete semigroups and for strongly left amenable semitopological semigroups.

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
Ralph DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 1139–1141. MR 159229, DOI 10.2140/pjm.1963.13.1139
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 156915
L. A. Karlovitz, On nonexpansive mappings, Proc. Amer. Math. Soc. 55 (1976), no. 2, 321–325. MR 405182, DOI 10.1090/S0002-9939-1976-0405182-X
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
Anthony To-ming Lau, Topological semigroups with invariant means in the convex hull of multiplicative means, Trans. Amer. Math. Soc. 148 (1970), 69–84. MR 257260, DOI 10.1090/S0002-9947-1970-0257260-5
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
Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630–641. MR 270356
Teck Cheong Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980), no. 1, 135–143. MR 599326, DOI 10.2140/pjm.1980.90.135
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
K. Salame, On the Schauder fixed point property, accepted for publication in Annals of Functional Analysis.
A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 111 (1935), no. 1, 767–776 (German). MR 1513031, DOI 10.1007/BF01472256