Invariant means and fixed point properties for nonexpansive representations of topological semigroups
type: Item
instance of: scholarly article
title: Invariant means and fixed point properties for nonexpansive representations of topological semigroups
zbMATH Open document ID: 0834.43001
DOI: 10.12775/TMNA.1995.003
published in: Topological Methods in Nonlinear Analysis
publication date: 1995-10-22T00:00:00Z
review text: Let \(S\) be a Hausdorff semitopological semigroup. A continuous bounded function \(f\) on \(S\) is called asymptotically uniformly continuous if for each \(s \in S\) and \(\varepsilon > 0\) there exist a neighbourhood \(U\) of \(s\) and a right ideal \(J\) of \(S\) such that for all \(u \in U\) \(\sup \{|f(ut) - f(st) |: t \in J\} < \varepsilon\). If \(S\) is left reversible, the set \(ALUC (S)\) of all such functions is a translation invariant algebra. A submean \(\mu\) on a linear subspace of \(\ell^\infty(S)\) which contains the constants satisfies \(\mu (f + g) \leq \mu (f) + \mu (g)\); \(\mu (\alpha f) = \alpha \mu (f)\) for positive scalars \(\alpha\); \(\mu (f) \leq \mu (g)\) if \(f \leq g\); and \(\mu (1) = 1\). \(S\) is called left subamenable if the space \(LUC (S)\) of left uniformly continuous functions on \(S\) has a left invariant submean. If \(S\) is left reversible, it is left subamenable; if \(S\) is discrete, the converse also holds. For subsemigroups of amenable groups, left amenability and left subamenability coincide. Let \(C\) be a subset of a Banach space \(E\). Let \(s \mapsto T_s\) be a representation of \(S\) as self-maps of the set \(C\). \(F(S)\) denotes the set of common fixed points of the family \(\{T_s : s \in S\}\). \(\{T_s\}\) is called left asymptotically nonexpansive if, given \(\varepsilon > 0\) and \(x \in C\), there is a left ideal \(J\) such that \(|T_sx - T_sy |\leq |x - y |+ \varepsilon\) for each \(s \in J\) and \(y \in C\). If this condition holds, \(E\) is uniformly convex, \(C\) is convex, and \(T_s\) is continuous for each \(s\), then \(F(S)\) is closed and convex. Under the same condition, for any subset \(C\) of any Banach space \(E\), if \(C\) contains an element \(z\) with bounded orbit, then the function \(f(t) = |T_tz - x |^2\) is asymptotically right uniformly continuous for each \(x \in E\). The authors go on to prove a number of fixed point theorems, of which the following is the simplest. If there is a left i…
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H20
zbMATH DE Number: 807913
zbMATH Keywords: Hausdorff semitopological semigroup
zbMATH Keywords: submean
zbMATH Keywords: left subamenable
zbMATH Keywords: left uniformly continuous functions
zbMATH Keywords: fixed points
zbMATH Keywords: fixed point theorems
zbMATH Keywords: left invariant submean
author: A. T.-M. Lau
author: Wataru Takahashi
reviewed by: J. Pym
MaRDI profile type: Publication
full work available at URL: tmna.1995.003
OpenAlex ID: W2575416291
description: scientific article; zbMATH DE number 807913