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Quasi-invariant lifts of completely positive maps for groupoid actions

Suvrajit Bhattacharjee, Marzieh Forough

TL;DR

This work develops a groupoid-generalized lifting theory for equivariant completely positive maps between $G$-C*-algebras. By combining $C_0(G^{(0)})$-nuclearity with averaging over quasi-invariant, quasi-central approximate units and the Busby invariant for groupoid actions, it establishes the existence of quasi-invariant CP lifts along $G$-extensions, with controlled asymptotic equivariance on compact subsets of the groupoid. The main result provides a substantial generalization of previous lifting theorems for groups to the groupoid setting and includes a construction of the Busby invariant in this context. The findings have implications for the analysis of dynamical systems indexed by groupoids and for equivariant KK-theory, connecting to broader lifting frameworks and upper semi-continuous field considerations.

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.

Quasi-invariant lifts of completely positive maps for groupoid actions

TL;DR

This work develops a groupoid-generalized lifting theory for equivariant completely positive maps between -C*-algebras. By combining -nuclearity with averaging over quasi-invariant, quasi-central approximate units and the Busby invariant for groupoid actions, it establishes the existence of quasi-invariant CP lifts along -extensions, with controlled asymptotic equivariance on compact subsets of the groupoid. The main result provides a substantial generalization of previous lifting theorems for groups to the groupoid setting and includes a construction of the Busby invariant in this context. The findings have implications for the analysis of dynamical systems indexed by groupoids and for equivariant KK-theory, connecting to broader lifting frameworks and upper semi-continuous field considerations.

Abstract

Let be a locally compact, Hausdorff, second countable groupoid and be a separable, -nuclear, --algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from into a separable, quotient -algebra. Along the way, we construct the Busby invariant for -actions.
Paper Structure (5 sections, 29 theorems, 102 equations)

This paper contains 5 sections, 29 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Groupoids and their actions
  3. Equivariant extensions and the Busby invariant
  4. Some technical results on quasi-central approximate units
  5. The main results

Key Result

Theorem 1.1

Let $G$ be a locally compact, Hausdorff, second countable group and $(A,\alpha)$ be a separable, nuclear, $G$-$C^*$-algebra. Let be a $G$-$C^*$-extension, with $B$ separable. Let $\psi : A \rightarrow D$ be a $G$-equivariant, completely positive and contractive map. Then there exists a sequence of completely positive and contractive maps $\varphi_n : A \rightarrow B$, $n \in \mathbb{N}$, such tha

Theorems & Definitions (68)

  • Theorem 1.1: forough-gardella-thomsen:lifting*Theorem 3.4
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2: kasparov:equivariant-kk*Definition 1.5
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7: le_gall:equivariant-kk*Definition 3.5
  • Lemma 2.8: muhly-williams:renault-theorem*Lemma 4.3
  • ...and 58 more