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

From previous work, we know how to obtain type II_0 E_0-semigroups using boundary weight doubles (ϕ, ν), where ϕ: M_n(\C) \to M_n(\C) is a unital q-positive map and νis a normalized unbounded boundary weight over L^2(0, \infty). In this paper, we classify the unital q-positive maps ϕ: M_2(\C) \to M_2(\C). We find that every unital q-pure map ϕ: 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_ϕfor all unital q-positive maps ϕ: M_3(\C) \to M_3(\C). In conclusion, we present a cocycle conjugacy result for E_0-semigroups induced by boundary weight doubles (ϕ, ν) when νhas the form ν(\sqrt{I - Λ(1)} B \sqrt{I - Λ(1)})=(f,Bf).