2010/07/26 by Christopher Jankowski, Jankowski, Christopher
Mathematics · #46L07 (Secondary) #46L57 (Primary) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46L07 #msc:46L57
paper · pdf · doi:10.48550/arxiv.1007.4459
20 pages
arxiv created 2010/07/26 · arxiv updated 2010/07/27
We have seen that if ϕ: Mn(\C) → Mn(\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 II0 E0-semigroup. Let ϕ: Mn(\C) → Mn(\C) and ψ: Mn'(\C) → Mn'(\C) be unital rank one q-positive maps, so for some states ρ∈ Mn(\C)^* and ρ' ∈ Mn'(\C)^*, we have ϕ(A)=ρ(A)In and ψ(D) = ρ'(D)In' for all A ∈ Mn(\C) and D ∈ Mn'(\C). We find that if νand ηare arbitrary type II Powers weights, then (ϕ, ν) and (ψ, η) induce non-cocycle conjugate E0-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(√(I - Λ(1)) B √(I - Λ(1)))=(f,Bf), then the E0-semigroups induced by (ϕ,ν) and (ψ, ν) are cocycle conjugate if and only if n=n' and ϕand ψare conjugate.