August 19, 2025

Quasi-invariant lifts of completely positive maps for groupoid actions

Key Points

Quasi-invariant, completely positive lifts exist for maps from A to a quotient C*-algebra, enhancing the understanding of groupoid actions.
Key evidence shows that these lifts maintain contractivity, opening new connections in the theory of maps and algebras.
The approach involves constructing the Busby invariant for G-actions, showcasing its significance in equivariant theory.
This may enable deeper insights into C*-algebra properties, as well as applications in broader mathematical fields.

Abstract

Abstract Let G be a locally compact, Hausdorff, second countable groupoid and A be a separable, C₀ (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.

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