Asymptotic lifting for completely positive maps

TL;DR

This paper develops a method to asymptotically lift completely positive maps between C*-algebras, preserving properties like linearity, positivity, and equivariance in the limit, with implications for understanding group amenability.

Contribution

It introduces a new asymptotic lifting technique for completely positive maps, including equivariant cases, and characterizes amenability via asymptotic equivariant sections.

Findings

Existence of asymptotic lifts with desired properties

Asymptotic equivariance under group actions

Characterization of amenability through asymptotic sections

Abstract

Let $A$ and $B$ be $C^*$-algebras with $A$ separable, let $I$ be an ideal in $B$, and let $\psi\colon A\to B/I$ be a completely positive contractive linear map. We show that there is a continuous family $\Theta_t\colon A\to B$, for $t\in [1,\infty)$, of lifts of $\psi$ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If $\psi$ is of order zero, then $\Theta_t$ can be chosen to have this property asymptotically. If $A$ and $B$ carry continuous actions of a second countable locally compact group $G$ such that $I$ is $G$-invariant and $\psi$ is equivariant, we show that the family $\Theta_t$ can be chosen to be asymptotically equivariant. If a linear completely positive lift for $\psi$ exists, we can arrange that $\Theta_t$ is linear and completely positive for all $t\in [1,\infty)$. In the equivariant setting, if $A$, $B$ and $\psi$ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if $G$ is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.