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

Abstract

Strongly self-absorbing ${\mathrm{C}}^{\ast }$-algebras play a distinguished role in the classification of nuclear ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-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.