$E_{0}^{P}$-semigroups and product systems

TL;DR

This paper studies $E_{0}^{P}$-semigroups associated with convex cones in $R^n$, demonstrating that their product systems are concrete and that the semigroups can be reconstructed from these systems up to cocycle conjugacy.

Contribution

It establishes that $E_{0}^{P}$-semigroups have concrete product systems and can be fully recovered from them, extending the theory to convex cones in higher dimensions.

Findings

Product systems associated with $E_{0}^{P}$-semigroups are concrete.

$E_{0}^{P}$-semigroups can be reconstructed from their product systems.

The results generalize existing theory to convex cones in $R^n$.

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 $\mathcal{E}_{\alpha}$