E_0-semigroups and q-purity

TL;DR

This paper characterizes q-pure E_0-semigroups of type II_0 derived from boundary weight maps with range rank one, and establishes the non-existence of such semigroups from rank two maps, providing classification criteria in finite dimensions.

Contribution

It describes all q-pure E_0-semigroups of type II_0 from boundary weight maps with range rank one and proves the non-existence from rank two, including classification criteria in finite dimensions.

Findings

All q-pure E_0-semigroups of type II_0 from rank one boundary weight maps are characterized.

No q-pure E_0-semigroups of type II_0 arise from rank two boundary weight maps.

A criterion is provided for cocycle conjugacy of rank one boundary weight maps in finite-dimensional cases.

Abstract

An E_0-semigroup is called q-pure if it is a CP-flow and its set of flow subordinates is totally ordered by subordination. The range rank of a positive boundary weight map is the dimension of the range of its dual map. Let K be a separable Hilbert space. We describe all q-pure E_0-semigroups of type II_0 which arise from boundary weight maps with range rank one over K. We also prove that no q-pure E_0-semigroups of type II_0 arise from boundary weight maps with range rank two over K. In the case when K is finite-dimensional, we provide a criterion to determine if two boundary weight maps of range rank one over K give rise to cocycle conjugate q-pure E_0-semigroups.