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# 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 $C^*$-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 $\mathbb R_{+}^k$.
 MSC Classifications: 47A20 - Dilations, extensions, compressions 46L08 - $C^*$-modules