Generalizing amenable group action classification to classifiable stably finite C*-algebras

Extend the Gabe–Szabó cocycle-conjugacy classification of outer actions of discrete amenable groups on Kirchberg algebras to actions on classifiable stably finite C*-algebras by determining appropriate invariants and establishing a complete classification up to cocycle conjugacy for such actions.

Background

The introduction notes that while Gabe and Szabó have classified outer actions of discrete amenable groups on Kirchberg algebras up to cocycle conjugacy using equivariant KK-theory, comparable results for stably finite, classifiable C*-algebras are not yet established. The paper positions the Razak–Jacelon algebra W as a monotracial analog of the injective II1 factor and focuses on classifying outer W-absorbing actions of finite groups on W, highlighting the K-theoretical challenges that arise for stably finite settings.

By achieving a complete classification for W-absorbing actions on W, the paper clarifies difficulties specific to stably finite contexts and suggests that broader extension to general classifiable stably finite C*-algebras remains an open direction. The cited work [8] provides the benchmark classification in the purely infinite (Kirchberg) case that motivates the open problem.