Lifting fixed points of completely positive semigroups

الأصل · EN

Let {ϕₛ}ₛ∈ ₛ be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra N. Assume there exists a semigroup {αₛ}ₛ∈ ₛ of weak*-continuous *-endomorphisms of some larger von Neumann algebra M⊃ N and a projection p∈ M with N=pMp such that αₛ(1-p)≤ 1-p for every s∈ S and ϕₛ(y)=pαₛ(y)p for all y∈ N. If ₛ∈ ₛαₛ(1-p)=0 then we show that the map E:M→ N defined by E(x)=pxp for x∈ M induces a complete isometry between the fixed point spaces of {αₛ}ₛ∈ ₛ and {ϕₛ}ₛ∈ ₛ.