Linearizability of non-expansive semigroup actions on metric spaces

We show that a non-expansive action of a topological semigroup S on a metric space X is linearizable iff its orbits are bounded. The crucial point here is to prove that X can be extended by adding a fixed point of S, thus allowing application of a semigroup version of the Arens–Eells linearization, iff the orbits of S in X are bounded.