Weakly almost periodic functions invariant means and fixed point properties in locally convex topological vector spaces. (English) Zbl 1517.47086

Salame, Khadime

Topol. Methods Nonlinear Anal. 60, No. 1, 135-152 (2022).

Let WAP\((S)\) denote the Banach algebra of weakly almost periodic functions on a given semitopological semigroup \(S\). During an International Conference on Fixed Point Theory which took place in Halifax in 1975, Anthony To-Ming Lau raised the question whether or not the left amenability property of WAP\((S)\) is equivalent to the existence of a common fixed point of any separately weakly continuous and weakly quasi-equicontinuous nonexpansive action of \(S\) on a weakly compact convex subset of a separated locally convex space. This remained an open problem for almost 50 years. In the present paper, the author gives an affirmative answer.

Reviewer: Jürgen Appell (Würzburg)

MSC:

47H10	Fixed-point theorems
46A03	General theory of locally convex spaces
43A07	Means on groups, semigroups, etc.; amenable groups
43A60	Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions

Keywords:

amenability; locally convex space; nonexpansive mapping; semigroup; (weakly) almost periodic function; weak topology

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