Lifts of completely positive (equivariant) maps

Eusebio Gardella  (Goteborgs Universitet)

16:00-17:00, November 15, 2021   Zoom Meeting 614 1453 0704 (Password: 132963)

Abstract:

Let $A$ and $B$ be $C^*$-algebras, $A$ separable and $I$ an ideal in $B$. We show that for any completely positive contractive linear map $\psi\colon A\to B/I$ 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 $A$ and $B$ carry continuous actions of a second countable locally compact group $G$ such that $I$ is $G$-invariant and $\psi$ is equivariant, then the family $\Theta_t$ can be chosen to be asymptotically equivariant. If a linear completely positive lift for $\psi$ exists, then we can arrange that $\Theta_t$ is linear and completely positive for all $t\in [1,\infty)$; this yields an equivariant version of the Choi-Effros lifting theorem. In the equivariant setting, if $A$, $B$ and $\psi$ are unital, the existence of asymptotically linear unital lifts are only guaranteed 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. This talk is based on joint work with Marzieh Forough and Klaus Thomsen.