On the existence of $E_0$-semigroups -- the multiparameter case

Let $P\subset {\mathbb{R}}^d$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P\cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P={\mathbb{R}}^d$. Denote the interior of $P$ by $\mathrm{\Omega }$. Let $E$ be a product system over $\mathrm{\Omega }$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $\alpha:={\alpha_x}_{x \in P}$ofunitalnormal$*$-endomorphismsof$B($\mathcal{H}$)$suchthat$E$ is isomorphic to the product system associated to $\alpha$.