Invariant means and fixed point properties for nonexpansive representations of topological semigroups
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 invariant submean on the space of asymptotically right uniformly continuous functions, \(E\) is a Hilbert space, \(C\) is closed and convex, and \(\{T_s\}\) is left asymptotically nonexpansive, then \(C\) contains a common fixed point for \(\{T_s\}\). The paper concludes with a discussion of the case in which \(E\) is a dual Banach space and \(C\) is convex and weak\(^*\) compact. Suppose that either \(LUC (S)\) is left amenable or \(S\) is discrete and \(LUC (S)\) is left subamenable. If \(\{T_s\}\) is norm nonexpansive and \((s,x) \mapsto T_s x\), \(S \times C \to C\) is jointly continuous when \(C\) has the weak\(^*\)-topology, then there is a common fixed point for \(\{T_s\}\) in \(C\).
scientific article; zbMATH DE number 807913