An essential representation for a product system over a finitely generated subsemigroup of ${\mathbb{Z}}^d$

Abstract

Let $S\subset {\mathbb{Z}}^d$ be a finitely generated subsemigroup. Let $E$ be a product system over $S$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $\alpha :=\{{\alpha }_x{\}}_{x\in S}$ of unital normal $\ast$-endomorphisms of $B(\mathcal{H})$ such that $E$ is isomorphic to the product system associated to $\alpha$.