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

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