Quasi-invariant lifts of completely positive maps for groupoid actions

Abstract

Let $G$ be a locally compact, Hausdorff, second countable groupoid and $A$ be a separable, $C_0(G^{(0)})$-nuclear, $G$-$C^{\ast }$-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 $C^{\ast }$-algebra. Along the way, we construct the Busby invariant for $G$-actions.

Final version

References in corpus (4)

• Renault's Equivalence Theorem for Groupoid Crossed Products
• Homology and K-theory of dynamical systems. I. torsion-free ample groupoids
• Equivariant ${\mathcal{O}}_2$-absorption theorem for exact groups
• The stable uniqueness theorem for equivariant Kasparov theory