Invariant submeans and semigroups of nonexpansive mappings on Banach spaces with normal structure (Q2563442)

A closed convex subset \(K\) of a Banach space has normal structure if for each closed convex subset \(H\) of \(K\) with more than one point there is \(x\in H\) with \(\sup \{|x-y |: y\in H\} < \text{diam} (H)\). A semigroup \(S\) is called left reversible if any two of its closed right ideals meet. A mean \(\mu\) on the space \(CB(S)\) of continuous bounded functions on a semitopological semigroup \(S\) is left subinvariant if, for all \(a\in S\), \(\mu (\ell_af) \geq\mu (f)\), where \(\ell_af\) is the left translate by \(a\) of the function \(f\) in \(CB(S)\). The main result of the paper is a generalisation of a theorem of \textit{T.-Ch. Lim} [Pac. J. Math. 53, 487-493 (1974; Zbl 0291.47032)]: Let \(S\) be a semitopological semigroup, let \(D\) be a nonempty weakly compact convex subset of a Banach space \(E\) which has normal structure, and let \({\mathcal S} = \{T_s: s\in S\}\) be a continuous representation of \(S\) as nonexpansive self mappings of \(D\). Suppose the space \(RUC(S)\) of right uniformly continuous bounded functions on \(S\) has a left invariant submean. Then \({\mathcal S}\) has a common fixed point in \(D\).