$E_0^P$-semigroups and product systems

Abstract

Let $P$ be a closed convex cone in ${\mathbb{R}}^n$. Assume that $P$ is spanning i.e. $P-P={\mathbb{R}}^n$ and pointed i.e. $P\cap -P=\{0\}$. Let $\alpha :=\{{\alpha }_x:x\in P\}$ be a $\sigma$-weakly continuous family of unital normal endomorphisms on $B(H)$. Denote the "product system" associated to $\alpha$ by ${\mathcal{E}}_{\alpha }$. We show that ${\mathcal{E}}_{\alpha }$ is a concrete product system and $\alpha$, up to cocycle conjugacy, can be recovered completely from ${\mathcal{E}}_{\alpha }$

14 Pages