Suvrajit Bhattacharjee: Equivariant lifting problem for continuous fields of completely positive maps

Abstract: Let \( G \) be a locally compact group, \(A\) and \(B\) be two \(G\)-\(C^{*}\)-algebras and \(J\) be a closed, two-sided, \(G\)-invariant ideal in \(B\). Let us write \(q : B \rightarrow B/J\) for the (\(G\)-equivariant) quotient map and consider the diagram \[ \begin{matrix} & & B \\ & \theta \nearrow & \downarrow q \\ A & \xrightarrow{\psi} & B/J \end{matrix} \] where \( \phi \) and \(\theta\) are completely positive, contractive, \(G\)-equivariant maps. The equivariant lifting problem for \(\phi\) asks whether one can find \(\theta\) so that the diagram commutes. In 1976, Choi and Effros proved that if \(G\) is the trivial group, \(A\) is separable and if \(\phi\) is a nuclear map, then the lifting problem for \(\phi\) can be solved. Inspired by Kasparov's equivariant \(\operatorname{KK}\)-theory, Forough-Gardella-Thomsen recently proved that one can always "average" out a not-necessarily-equivariant lift \(\theta\) to obtain a sequence of lifts \( (\theta_{n}) \) that are asymptotically \(G\)-equivariant, i.e., \[ \lim_{n \rightarrow \infty}\left\Vert \theta_{n}(ga)-g\theta_{n}(a) \right\Vert=0, \] for all \(g \in G\) and \(a \in A\). The purpose of the talk is to present an extension of this result to the realm of groupoid actions. Although we will present the main lifting result for arbitrary locally compact, Hausdorff groupoid actions, we will mostly talk about a particular case, namely, Rieffel's (upper-semi-) continuous fields of \(C^{*}\)-algebras coming from group actions, or more generally, Kasparov's \(G\)-\(C_{0}(X) \)-algebras, where, as before, \(G\) is a locally compact group. This talk is based on an ongoing work with Marzieh Forough.