Unital q-positive maps on M_2(\C) and a related E_0-semigroup result

TL;DR

This paper classifies unital q-positive maps on 2x2 matrices, explores their properties for 3x3 matrices, and discusses implications for related E_0-semigroups, advancing understanding of their structure and classification.

Contribution

It provides a complete classification of unital q-positive maps on M_2(C) and analyzes their limits on M_3(C), contributing to the theory of E_0-semigroups.

Findings

Every unital q-pure map on M_2(C) is either rank one or invertible.

The limit maps for unital q-positive maps on M_3(C) are characterized.

A cocycle conjugacy result for E_0-semigroups induced by boundary weight doubles is established.

Abstract

From previous work, we know how to obtain type II_0 E_0-semigroups using boundary weight doubles (\phi, \nu), where \phi: M_n(\C) \to M_n(\C) is a unital q-positive map and \nu is a normalized unbounded boundary weight over L^2(0, \infty). In this paper, we classify the unital q-positive maps \phi: M_2(\C) \to M_2(\C). We find that every unital q-pure map \phi: M_2(\C) \to M_2(\C) is either rank one or invertible. We also examine the case n=3, finding the limit maps L_\phi for all unital q-positive maps \phi: M_3(\C) \to M_3(\C). In conclusion, we present a cocycle conjugacy result for E_0-semigroups induced by boundary weight doubles (\phi, \nu) when \nu has the form \nu(\sqrt{I - \Lambda(1)} B \sqrt{I - \Lambda(1)})=(f,Bf).