Lifting fixed points of completely positive semigroups

Abstract

Let $\{\phi_s\}_{s\in S}$ 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 $M\supset N$ and a projection $p\in M$ with $N=pMp$ such that $\alpha_s(1-p)\le 1-p$ for every $s\in S$ and $\phi_s(y)=p\alpha_s(y)p$ for all $y\in N$. If $\inf_{s\in S}\alpha_s(1-p)=0$ then we show that the map $E:M\to N$ defined by $E(x)=pxp$ for $x\in M$ induces a complete isometry between the fixed point spaces of $\{\alpha_s\}_{s\in S}$ and $\{\phi_s\}_{s\in S}$.