A family of non-cocycle conjugate E₀-semigroups obtained from boundary weight doubles

Let ρ ∈ M_n(C)* and ρ′ ∈ M_n′ (C)* be states, and define unital q-positive maps φ and ψ by φ(A) = ρ(A)I_n and ψ(D) = ρ′(D)I_n′ for all A ∈ M_n (C) and D ∈ M_n′ (C). We show that if v and η are type II Powers weights, then the boundary weight doubles (φ, v) and (ψ, η) induce non-cocycle con¬jugate E₀-semigroups if ρ and ρ′ have different eigenvalue lists. We then classify the q-corners and hyper maximal q-corners from φ to ψ, finding that if v is a type II Powers weight of the form where (l) ∈ B(L²(0,∞)) is the operator of multiplication by e^-x, then the E₀-semigroups induced by (φ v) and (ψ, v) are cocycle conjugate if and only if n = n′ and φ and ψ are conjugate.