Paper detail

A family of non-cocycle conjugate E_0-semigroups obtained from boundary weight doubles

We have seen that if ϕ: M_n(\C) \rightarrow M_n(\C) is a unital q-positive map and νis a type II Powers weight, then the boundary weight double (ϕ, ν) induces a unique (up to conjugacy) type II_0 E_0-semigroup. Let ϕ: M_n(\C) \rightarrow M_n(\C) and ψ: M_{n'}(\C) \rightarrow M_{n'}(\C) be unital rank one q-positive maps, so for some states ρ\in M_n(\C)^* and ρ' \in M_{n'}(\C)^*, we have ϕ(A)=ρ(A)I_n and ψ(D) = ρ'(D)I_{n'} for all A \in M_n(\C) and D \in M_{n'}(\C). We find that if νand ηare arbitrary type II Powers weights, then (ϕ, ν) and (ψ, η) induce non-cocycle conjugate E_0-semigroups if ρand ρ' have different eigenvalue lists. We then completely classify the q-corners and hyper maximal q-corners from ϕto ψ, obtaining the following result: If νis a type II Powers weight of the form ν(\sqrt{I - Λ(1)} B \sqrt{I - Λ(1)})=(f,Bf), then the E_0-semigroups induced by (ϕ,ν) and (ψ, ν) are cocycle conjugate if and only if n=n' and ϕand ψare conjugate.