An application of generalised limits to obtain continuous- from discrete time dilations

Ptak (Ann. Polon. Math. 45(3), 237–243 (1985)) introduced an elegant method in the separable setting that allows us to deduce dilation results for continuous semigroups of contractions from the corresponding discrete versions. The purpose of this note is to record another such trick using generalised limits that shows that dilation results for semigroups of contractions in continuous time come for free once we establish them in discrete time. Moreover, it works for non-separable Hilbert spaces and bypasses the theory of unbounded operators.