Isometric Dilations of Representations of Product Systems via Commutants

We construct a weak dilation of a not necessarily unital CP-semigroup to an E-semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E-semigroup has a pre-assigned product system. Then, making use of the commutant of von Neumann correspondences, we apply the dilation theorem to proof that covariant representations of product systems admit isometric dilations.Comment: Switched some definitions directly after theorems in Sect. 1, included (>6 pp.) introduction to von Neumann correspondences (Sect. 3). To appear in International Journal of Mathematic