Quasi-invariant lifts of completely positive maps for groupoid actions

Let \(G\) be a locally compact, \(\sigma\)-compact, Hausdorff groupoid and \(A\) be a separable, \(C_0(G^{(0)})\)-nuclear, \(G\)-\(C^*\)-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from \(A\) into a separable, quotient \(C^*\)-algebra. Along the way, we construct the Busby invariant for \(G\)-actions.