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 $\psi :A\to B/I$ be a completely positive contractive linear map. We show that there is a continuous family ${\Theta }_t:A\to B$, for $t\in [1,\infty )$, of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ 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 ψ is equivariant, we show that the family ${\Theta }_t$ can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ 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 ψ 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.