Unital q-positive maps on M₂(C) and a related E₀-semigroup result

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).