Lifts of completely positive (equivariant) maps

Eusebio Gardella  (Goteborgs Universitet)

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


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.

About the speaker: