2021/03/16 by Forough, Marzieh, Gardella, Eusebio, Thomsen, Klaus · 1 citation
#FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2103.09176
Let A and B be C^*-algebras with A separable, let I be an ideal in B, and let ψ\colon A→ B/I be a completely positive contractive linear map. We show that there is a continuous family Θt\colon A→ B, for t∈ [1,∞), of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then Θt 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 Θt can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that Θt is linear and completely positive for all t∈ [1,∞). 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.