Additive units of product system

TL;DR

This paper introduces additive units and roots in spatial product systems, revealing their structure, and connects these concepts to Powers' problem and random set theory, advancing the understanding of product system classification.

Contribution

It defines additive units and roots, shows their relation to the system's index, and offers an alternative proof of Powers' problem, linking product subsystems to random sets.

Findings

Roots form a Hilbert space with dimension equal to the system's index.

Units and roots generate the type I part of the product system.

Established a connection between product subsystem clusters and random sets.

Abstract

We introduce the notion of additive units and roots of a unit in a spatial product system. The set of all roots of any unit forms a Hilbert space and its dimension is the same as the index of the product system. We show that a unit and all of its roots generate the type I part of the product system. Using properties of roots, we also provide an alternative proof of the Powers' problem that the cocycle conjugacy class of Powers sum is independent of the choice of intertwining isometries. In the last section, we introduce the notion of cluster of a product subsystem and establish its connection with random sets in the sense of Tsirelson and Liebscher.