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Toward the classification of strongly self-absorbing $\mathrm{C}^*$-dynamical systems of compact groups

Masaki Izumi, Keiya Ohara

Abstract

Strongly self-absorbing $\mathrm{C}^*$-algebras play a distinguished role in the classification of nuclear $\mathrm{C}^*$-algebras. Their dynamical analogues were introduced and extensively studied by Szabó. In this paper, we propose a conjecture regarding the equivariant $KK$-theory of strongly self-absorbing $\mathrm{C}^*$-dynamical systems of compact groups in the equivariant bootstrap category; an affirmative answer to this conjecture would lead to classification results. We settle this conjecture for all finite EPPO (every element has a prime-power order) groups. In the course of our proof, we establish a Künneth-type formula for the equivariant $K$-theory of $\mathrm{C}^*$-algebras equipped with finite cyclic group actions -- more precisely, for the cyclotomic part of the equivariant $K$-groups introduced by Meyer and Nadareishvili -- under a certain unique divisibility assumption.

Toward the classification of strongly self-absorbing $\mathrm{C}^*$-dynamical systems of compact groups

Abstract

Strongly self-absorbing -algebras play a distinguished role in the classification of nuclear -algebras. Their dynamical analogues were introduced and extensively studied by Szabó. In this paper, we propose a conjecture regarding the equivariant -theory of strongly self-absorbing -dynamical systems of compact groups in the equivariant bootstrap category; an affirmative answer to this conjecture would lead to classification results. We settle this conjecture for all finite EPPO (every element has a prime-power order) groups. In the course of our proof, we establish a Künneth-type formula for the equivariant -theory of -algebras equipped with finite cyclic group actions -- more precisely, for the cyclotomic part of the equivariant -groups introduced by Meyer and Nadareishvili -- under a certain unique divisibility assumption.
Paper Structure (12 sections, 67 theorems, 175 equations)

This paper contains 12 sections, 67 theorems, 175 equations.

Table of Contents

  1. Acknowledgements.
  2. Preliminaries
  3. $K$-theory and $KK$-theory
  4. Some remarks on ring theory and homological algebra
  5. Strongly self-absorbing actions
  6. Model actions
  7. Equivariant Künneth formula
  8. $K$-theory of strongly self-absorbing actions
  9. Actions of cyclic $p$-groups
  10. Actions of EPPO-groups
  11. Proof of Theorem \ref{['EPPO family']}
  12. The case of the Lie groups $U(1)$ and $SU(2)$

Key Result

Theorem 2

Conjecture A is true for every EPPO-group.

Theorems & Definitions (154)

  • Definition 1
  • Conjecture 1
  • Remark 2
  • Definition 3
  • Example 4
  • Theorem 2: Theorem \ref{['EPPO case']}
  • Theorem 3: Theorem \ref{['Kunneth']}
  • Definition 1.1: cf. Blackadar
  • Definition 1.2: cf. Kasparov
  • Proposition 1.3: cf. Kasparov
  • ...and 144 more