Free groups are shown to admit amenable actions on unital simple AF-algebras, closing the last open case for free groups

Free groups admit amenable actions on unital simple AF-algebras. Within the existence problem for amenable actions of non-amenable groups on classifiable simple C*-algebras, free groups had remained an unresolved case; this result fills that gap. The abstract states the conclusion directly with "We show" and notes that the proof relies on the Baire category theorem; the text provides no proof details, construction steps, or verification data.

The result is presented as settling, for free groups, the last remaining case of the existence problem for amenable actions of non-amenable groups on classifiable simple C*-algebras. It moves the free-group case from open to resolved, making the picture of that existence problem complete in the free-group direction. Based on the abstract's phrase "settles, for free groups, the last remaining case"; this is the authors' positioning of the problem's status, and the text does not list the other group cases or comparisons.

The proof relies on the Baire category theorem. It supplies a specific proof route to the existence conclusion rather than only asserting that a construction exists. The abstract explicitly writes "The proof relies on the Baire category theorem"; it does not elaborate on how the theorem is applied.

The result was obtained with substantial assistance from GPT-6 Astra. It discloses, at the abstract level, the involvement of an AI system in the mathematical research process. The abstract states "This result was obtained with substantial assistance from GPT-6 Astra"; the text does not specify the scope or division of that assistance.

The result concerns the existence problem for amenable actions of non-amenable groups on classifiable simple C*-algebras and applies to the free-group case; for researchers tracking whether that problem is fully covered, it provides the answer in the free-group direction. The abstract notes that the proof relies on the Baire category theorem, so its setting is theoretical argument around that existence problem rather than a constructive algorithm or numerical experiment.

Only the abstract is available, so the concrete construction in the proof, the way the Baire category theorem is applied, and the list of prior cases underlying the phrase "the last remaining case" are not given in the text; the full details of the proof therefore cannot be judged from the abstract. The abstract mentions that the result was obtained with substantial assistance from GPT-6 Astra but does not describe the scope or manner of that assistance, which remains open information for understanding the process.