Linearizability of non-expansive semigroup actions on metric spaces (Q935259)

The author shows that a non-expansive action of a topological semigroup \(S\) on a metric space \(X\) is linearizable if and only if each of its orbits is bounded. (The action is said to be linearizable if it can be isometrically realized by restricting an action of \(S\) by linear contractions on a normed space \(V\) to a metric subspace). If there is a fixed point, then one can do this by a semigroup version of the Arens-Eels isometric embedding of a metric space into a normed space. Thus the crucial point in the proof is showing in the absence of a fixed point that the action can be extended by adding a fixed point.