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A universal coefficient theorem for actions of finite groups on C*-algebras

Ralf Meyer, George Nadareishvili

TL;DR

The paper advances the dynamical classification program for finite-group actions on C*-algebras by proving a Universal Coefficient Theorem for the $G$-equivariant bootstrap class after inverting the group order $|G|$, and showing this bootstrap class is generated by $C(G/H)$ with $H$ cyclic via the Arano–Kubota result. It develops a localisation framework at sets of primes and then at the group order, introducing $S$-divisible objects and a Mackey/Grothendieck-type invariant that reduces KK^G-objects to modules over explicit rings $bZ[ heta_n,1/|G|] times W_H$. The main contribution is a computable UCT and a classification of certain outer $G$-actions on stable Kirchberg algebras up to cocycle conjugacy, expressed through a hereditary abelian target category $ rak A_G$ and a universal invariant. This work connects equivariant KK-theory, localisation techniques, and dynamical Kirchberg–Phillips theory to enable concrete dynamical classification in the finite-group setting.

Abstract

The equivariant bootstrap class in the Kasparov category of actions of a finite group G consists of those actions that are equivalent to one on a Type I C*-algebra. Using a result by Arano and Kubota, we show that this bootstrap class is already generated by the continuous functions on G/H for all cyclic subgroups H of G. Then we prove a Universal Coefficient Theorem for the localisation of this bootstrap class at the group order |G|. This allows us to classify certain G-actions on stable Kirchberg algebras up to cocycle conjugacy.

A universal coefficient theorem for actions of finite groups on C*-algebras

TL;DR

The paper advances the dynamical classification program for finite-group actions on C*-algebras by proving a Universal Coefficient Theorem for the -equivariant bootstrap class after inverting the group order , and showing this bootstrap class is generated by with cyclic via the Arano–Kubota result. It develops a localisation framework at sets of primes and then at the group order, introducing -divisible objects and a Mackey/Grothendieck-type invariant that reduces KK^G-objects to modules over explicit rings . The main contribution is a computable UCT and a classification of certain outer -actions on stable Kirchberg algebras up to cocycle conjugacy, expressed through a hereditary abelian target category and a universal invariant. This work connects equivariant KK-theory, localisation techniques, and dynamical Kirchberg–Phillips theory to enable concrete dynamical classification in the finite-group setting.

Abstract

The equivariant bootstrap class in the Kasparov category of actions of a finite group G consists of those actions that are equivalent to one on a Type I C*-algebra. Using a result by Arano and Kubota, we show that this bootstrap class is already generated by the continuous functions on G/H for all cyclic subgroups H of G. Then we prove a Universal Coefficient Theorem for the localisation of this bootstrap class at the group order |G|. This allows us to classify certain G-actions on stable Kirchberg algebras up to cocycle conjugacy.
Paper Structure (8 sections, 26 theorems, 36 equations)

This paper contains 8 sections, 26 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Equivariant KK-theory
  3. Functors on KKG
  4. Homological algebra in KKG
  5. Generators for the equivariant bootstrap class
  6. Localisation
  7. Localisation at a set of primes
  8. Localisation at the group order

Key Result

Theorem 1.1

Let $G$ be a finite group. Let $A$ and $B$ be $G$-$\textup{C}^*$-algebras. Suppose that $A$ is in the $G$-equivariant bootstrap class, that is, it is $\mathrm{KK}^G$-equivalent to an action on a Type I $\textup{C}^*$-algebra. Suppose that $B$ is $\lvert G\rvert$-divisible in the sense that multiplic Here the products run over conjugacy classes of cyclic subgroups $H\subseteq G$. If $M_H$ are count

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 2.1: Arano-Kubota:Atiyah-Segal*Corollary 3.13.(1)
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Example 2.4
  • Remark 2.5
  • Theorem 2.6: Dell'Ambrogio dellAmbrogio:Cell_G*Theorem 4.9
  • Theorem 3.1
  • ...and 45 more