Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability

Every action of a countable discrete amenable group on a simple, separable, unital, infinite-dimensional, nuclear, stably finite complex C^(∗)-algebra that is already Jiang–Su stable absorbs the trivial action on the Jiang–Su algebra up to cocycle conjugacy. No restriction is imposed on the action on the tracial-state simplex. This proves the unital, stably finite case of Szabó's Conjecture A on automatic equivariant Jiang–Su stability.

We prove that strict comparison in the extended-functional sense implies Jiang–Su absorption for separable simple nuclear non-elementary C^(∗)-algebras. This resolves the corresponding implication of the Toms–Winter regularity problem, including nonunital algebras and allowing unbounded traces. More generally, every separable nuclear C^(∗)-algebra whose Cuntz semigroup is almost unperforated and fully almost divisible absorbs the Jiang–Su algebra.

For separable nuclear C^(∗)-algebras with no nonzero elementary ideal subquotients, finite nuclear dimension is equivalent to Jiang–Su stability. More generally, \dim_{nuc}(A_0\otimes\mathcal Z)\le1 for every separable nuclear A_(0). This proves Robert and Tikuisis's Conjecture (C1) and the nuclear-dimension equivalence in the nonsimple Toms–Winter regularity question.

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 uniform property Γ. This answers Problem XXI of Schafhauser, Tikuisis and White affirmatively. Independently, strict comparison implies Jiang–Su absorption for simple, separable, unital, infinite-dimensional nuclear algebras, resolving the unital Toms–Winter conjecture. Under comparison tested on traces of finite target rank, we also obtain uniform property Γ and absorption for simple, separable, nuclear, stably projectionless algebras whose densely finite traces are all bounded and have a nonempty compact normalized base.