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The stable uniqueness theorem for equivariant Kasparov theory

James Gabe, Gábor Szabó

TL;DR

The paper extends the stable uniqueness framework of $KK$-theory to the equivariant setting via the Cuntz–Thomsen picture, proving that anchored $(\alpha,\beta)$-Cuntz pairs with zero $KK^G$-class are stably operator homotopic. It develops a robust toolkit—operator homotopy, absorption, and asymptotic unitary equivalence—within the dynamical context, leveraging Kasparov-type results and quasicentral approximate units. The main achievement, the dynamical stable uniqueness theorem, provides equivalent characterizations of equivalence for cocycle representations and underpins future classifications of amenable group actions on Kirchberg algebras by equivariant $KK$-theory. This establishes a concrete, versatile foundation for dynamical classification problems in noncommutative geometry and operator algebras.

Abstract

This paper examines and strengthens the Cuntz-Thomsen picture of equivariant Kasparov theory for arbitrary second-countable locally compact groups, in which elements are given by certain pairs of cocycle representations between C*-dynamical systems. The main result is a stable uniqueness theorem that generalizes a fundamental characterization of ordinary $KK$-theory by Lin and Dadarlat-Eilers. Along the way, we prove an equivariant Cuntz-Thomsen picture analog of the fact that the equivalence relation of homotopy agrees with the (a priori stronger) equivalence relation of stable operator homotopy. The results proved in this paper will be employed as the technical centerpiece in forthcoming work of the authors to classify certain amenable group actions on Kirchberg algebras by equivariant Kasparov theory.

The stable uniqueness theorem for equivariant Kasparov theory

TL;DR

The paper extends the stable uniqueness framework of -theory to the equivariant setting via the Cuntz–Thomsen picture, proving that anchored -Cuntz pairs with zero -class are stably operator homotopic. It develops a robust toolkit—operator homotopy, absorption, and asymptotic unitary equivalence—within the dynamical context, leveraging Kasparov-type results and quasicentral approximate units. The main achievement, the dynamical stable uniqueness theorem, provides equivalent characterizations of equivalence for cocycle representations and underpins future classifications of amenable group actions on Kirchberg algebras by equivariant -theory. This establishes a concrete, versatile foundation for dynamical classification problems in noncommutative geometry and operator algebras.

Abstract

This paper examines and strengthens the Cuntz-Thomsen picture of equivariant Kasparov theory for arbitrary second-countable locally compact groups, in which elements are given by certain pairs of cocycle representations between C*-dynamical systems. The main result is a stable uniqueness theorem that generalizes a fundamental characterization of ordinary -theory by Lin and Dadarlat-Eilers. Along the way, we prove an equivariant Cuntz-Thomsen picture analog of the fact that the equivalence relation of homotopy agrees with the (a priori stronger) equivalence relation of stable operator homotopy. The results proved in this paper will be employed as the technical centerpiece in forthcoming work of the authors to classify certain amenable group actions on Kirchberg algebras by equivariant Kasparov theory.
Paper Structure (5 sections, 25 theorems, 207 equations)

This paper contains 5 sections, 25 theorems, 207 equations.

Table of Contents

  1. Preliminaries
  2. From homotopy to stable operator homotopy
  3. Absorption of cocycle representations
  4. Criteria for asymptotic unitary equivalence
  5. The dynamical stable uniqueness theorem

Key Result

Theorem 1

Let $A$ and $B$ be separable $\mathrm{C}^*$-algebras, and let $G$ be a second-countable locally compact group. Let $\alpha: G\curvearrowright A$ and $\beta: G\curvearrowright B$ be two continuous actions. Let be a pair of cocycle representations forming an anchored $(\alpha, \beta)$-Cuntz pair (see def:anchored). Then $[ (\varphi, \mathbbmss{u}), (\psi, \mathbbmss{v})] = 0$ in $KK^G(\alpha, \beta

Theorems & Definitions (65)

  • Theorem
  • Definition 1: see Szabo21cc
  • Remark 1
  • Definition 2: cf. Thomsen98
  • Definition 3: cf. Thomsen98
  • Definition 4: see Thomsen98
  • Proposition 1: cf. JensenThomsen
  • proof
  • Lemma 1
  • proof
  • ...and 55 more