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

Abstract

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 $\ast$-endomorphisms of $B(\mathcal{H})$ such that $E$ is isomorphic to the product system associated to $\alpha$.

arXiv admin note: text overlap with arXiv:1709.08152, Title changed, typos corrected and minor corrections