Representing a product system representation as a contractive semigroup and applications to regular isometric dilations

In this paper we propose a new technical tool for analyzing representations of Hilbert $C^{\ast }$-product systems. Using this tool, we give a new proof that every doubly commuting representation over ${\mathbb{N}}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of ${\mathbb{R}}_{+}^k$.