Unital q-positive maps on M₂(C) and cocycle conjugacy of E ₀-semigroups

From previous work, we know how to obtain type II₀ E ₀-semigroups using boundary weight doubles (φ, ν), where φ : M_n(C) → M_n(C) is a unital q-positive map and ν is a normalized unbounded boundary weight over L²(0, _∞). In this paper, we classify the unital q-positive maps φ : M₂(C) → M₂(C). We find that every unital q-pure map φ : M₂(C) → M₂(C) is either rank one or invertible. We also examine the case n = 3, finding the limit maps L _φ for all unital q-positive maps φ : M₃(C) → M₃(C). In conclusion, we present a cocycle conjugacy result for E₀-semigroups induced by boundary weight doubles (φ, ν) when ν has the form ν {equation presented}.