Lifting fixed points of completely positive semigroups

TL;DR

This paper establishes conditions under which fixed points of a semigroup of completely positive maps on a von Neumann algebra can be lifted to a larger algebra, preserving their structure via a complete isometry.

Contribution

It introduces a lifting theorem for fixed points of semigroups of completely positive maps using dilations to larger von Neumann algebras.

Findings

Fixed point spaces are completely isometric under certain dilation conditions.

The existence of a dilation with a projection p satisfying specific properties is crucial.

The result generalizes fixed point lifting in the context of von Neumann algebras.

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 $\{\phi_s\}_{s\in S}$.