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

TL;DR

This paper advances the classification of strongly self-absorbing C*-dynamical systems with compact group actions by proposing and confirming a conjecture in equivariant KK-theory, including new formulas for equivariant K-theory under specific conditions.

Contribution

It introduces a conjecture on equivariant KK-theory for these systems and proves it for finite EPPO groups, also establishing a new Kunneth-type formula for equivariant K-theory.

Findings

Confirmed the conjecture for all finite EPPO groups.

Established a Kunneth-type formula for equivariant K-theory with finite cyclic group actions.

Provided tools for classification of strongly self-absorbing C*-dynamical systems.

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 Szab\'o. 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 K\"unneth-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.