Asymptotic lifting for completely positive maps

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.