  In this paper we propose a new technical tool for analyzing representations
of Hilbert $C^*$-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 $\mathbb{R}_+^k$.
