On the existence of E₀-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 $\Omega$. Let $E$ be a product system over $\Omega$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $\alpha:=\{\alpha_x\}_{x \in P}$ of unital normal $*$-endomorphisms of $B(\mathcal{H})$ such that $E$ is isomorphic to the product system associated to $\alpha$.