Lifting fixed points of completely positive semigroups

Let $\{{\varphi }_s{\}}_{s\in S}$ 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 $\{{\alpha }_s{\}}_{s\in S}$ of weak*-continuous *-endomorphisms of some larger von Neumann algebra $M\supset N$andaprojection$p\in M$with$N=pMp$suchthat$\alpha_s(1-p)\le 1-p$ for every $s\in S$ and ${\varphi }_s(y)=p{\alpha }_s(y)p$ for all $y\in N$. If $\inf_{s\in S}\alpha_s(1-p)=0$thenweshowthatthemap$E:M\to N$definedby$E(x)=pxp$for$x\in M$inducesacompleteisometrybetweenthefixedpointspacesof${\alpha_s}{s\in S}$and${\phi_s}{s\in S}$.