Quasi-invariant lifts of completely positive maps for groupoid actions

TL;DR

This paper develops a framework for lifting equivariant completely positive maps in the setting of groupoid actions on C*-algebras, introducing quasi-invariant lifts and constructing the Busby invariant.

Contribution

It introduces the concept of quasi-invariant lifts for equivariant maps and constructs the Busby invariant for groupoid actions, advancing the theory of C*-algebraic groupoid actions.

Findings

Existence of quasi-invariant, completely positive, contractive lifts.

Construction of the Busby invariant for G-actions.

Framework applicable to separable, G-C*-algebras.

Abstract

Let $G$ be a locally compact, Hausdorff, second countable 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.