Lifting fixed points of completely positive semigroups

Abstract

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 $M\supset N$ and a projection $p\in M$ with $N=pMp$ such that ${\alpha }_s(1-p)\le 1-p$ for every $s\in S$ and ${\varphi }_s(y)=p{\alpha }_s(y)p$ for all $y\in N$. If ${\inf }_{s\in S}{\alpha }_s(1-p)=0$ 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 $\{{\varphi }_s{\}}_{s\in S}$.

4 pages, a new theorem is added, showing that part 3 of Theorem 1 holds for any commutative semigroup