Uniform property Gamma from real rank zero of tracial ultrapowers

For a simple separable unital infinite-dimensional nuclear stably finite C*-algebra with traces, does real rank zero in the uniform tracial ultrapower of its uniform tracial completion imply uniform property Γ?

Why it matters and what counts as progress

Why it matters: The question tests whether tracial projection approximations force the central divisibility identities missing from earlier results.

What counts as progress: Central projections halving every specified trace on every coefficient for each chosen free ultrafilter, without adding hypotheses beyond the stated algebra class.

How the problem developed

12 Jun 2025 · Open problem [1]
Christopher Schafhauser, Aaron Tikuisis and Stuart White Problem XXI asks whether real rank zero of the uniform tracial ultrapower forces uniform property Γ.
23 Sept 2026 · Reported claim [2]
OpenAI Reported an affirmative result for simple, separable, unital, infinite-dimensional, nuclear, stably finite C*-algebras with traces; the linked Lean work formalizes the principal implication in its stated setting.

AI contribution

Tracial projection methods and uniform property Gamma

OpenAI

For a simple separable unital infinite-dimensional nuclear stably finite C*-algebra with traces, real rank zero of the uniform tracial ultrapower of its uniform tracial completion implies a central projection halving every specified trace on every coefficient. This yields uniform property Gamma for each specified free ultrafilter.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4 theorem with Comparator statement alignment)

Human understandingFew