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

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