1 · The essentials
This family of manuscripts studies when adding a particular mathematical building block leaves an algebra unchanged in the relevant sense. The building block is called the Jiang–Su algebra. The main manuscript considers algebras that already have this property and asks whether it survives when a group acts on them. It shows in the manuscript that the answer is yes for the specified class of algebras and groups, with the precise conditions listed below. Importantly, it places no extra restriction on how the group moves the algebra’s traces. Companion manuscripts connect this stability with ways of comparing elements and measuring approximation complexity. The family includes a Lean formalization, identified in the source with the companion manuscript on tracial projection methods rather than with every assertion in the family.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a workshop that can incorporate an extra standard workbench without changing how it functions. The main question is whether this remains possible while respecting a whole collection of coordinated rearrangements of the workshop, not merely its stationary layout.
Why it matters
The main manuscript resolves the stated unital, stably finite case of Szabó’s Conjecture A without restrictions on trace dynamics. The family also establishes in the manuscripts the unital Toms–Winter equivalence between strict comparison, finite nuclear dimension, and Jiang–Su stability. The companion results extend selected parts of this connection beyond the simple unital setting, under their own hypotheses.
2 · Going further
Let be a simple, separable, unital, infinite-dimensional, nuclear, stably finite complex -algebra that is Jiang–Su stable, and let be a countable discrete amenable group. The main manuscript shows that every action of on absorbs the trivial action on the Jiang–Su algebra up to cocycle conjugacy. There is no restriction on the induced action on the tracial-state simplex. This is the unital, stably finite case of Szabó’s Conjecture A identified in the source.
The companion comparison manuscript shows that strict comparison in the extended-functional sense implies Jiang–Su absorption for separable, simple, nuclear, non-elementary -algebras. Its scope includes nonunital algebras and unbounded traces. It also gives a more general sufficient condition: a separable nuclear algebra absorbs when its Cuntz semigroup is almost unperforated and fully almost divisible.
The nuclear-dimension manuscript shows that finite nuclear dimension is equivalent to Jiang–Su stability for separable nuclear algebras with no nonzero elementary ideal subquotients. It also states the bound for every separable nuclear . The source identifies these results with Robert and Tikuisis’s Conjecture (C1) and the nuclear-dimension equivalence in the nonsimple Toms–Winter regularity question.
The abstracts describe a network of implications, not the detailed argument for the action theorem. The tracial projection companion supplies a specific route: real rank zero of the uniform tracial ultrapower of the uniform tracial completion implies uniform property , under its stated simple, separable, unital, infinite-dimensional, nuclear, stably finite hypotheses and the presence of traces. It independently establishes the unital strict-comparison-to-absorption implication and gives a further result for certain stably projectionless algebras. This companion is the manuscript marked Lean-formalized in the source; the supplied material does not specify a theorem-by-theorem formalization boundary.
Useful notions
- C*-algebras and group actions
- Tensor products and absorption
- Traces and comparison
- Nuclear dimension
- Uniform tracial constructions
What was known before / what is new
Before
The source presents automatic equivariant Jiang–Su stability as the subject of Szabó’s Conjecture A. It presents the relationships among comparison, nuclear dimension, and Jiang–Su absorption as the Toms–Winter regularity problem. It also names Robert and Tikuisis’s Conjecture (C1) and Problem XXI of Schafhauser, Tikuisis and White, but gives no further historical account of prior partial results.
New
The main manuscript shows automatic equivariant Jiang–Su stability in the stated unital, stably finite amenable-action setting, without a trace-dynamics restriction. The comparison and nuclear-dimension companions establish the stated implications and equivalences, including extensions to nonunital or nonsimple settings under additional conditions. The Lean-marked tracial projection manuscript answers the named Problem XXI affirmatively and includes the unital comparison-to-absorption result.
Limits and assumptions
The action theorem requires a countable discrete amenable group and an already Jiang–Su-stable algebra satisfying every listed hypothesis; it does not assert automatic absorption for arbitrary actions or arbitrary algebras. The nonsimple nuclear-dimension equivalence requires no nonzero elementary ideal subquotients. The stably projectionless result requires comparison tested on traces of finite target rank, with all densely finite traces bounded and possessing a nonempty compact normalized base, as well as simplicity, separability, and nuclearity. Verification is present in this family: Manuscript 4 is explicitly marked Lean-formalized, but the source does not identify every formalized claim or establish Lean coverage of the action theorem. The source resolves the specified cases and does not state the status of the remaining cases of the broader conjectures.
Mini glossary
- Jiang–Su stability —
- The absorption property for the Jiang–Su algebra studied throughout this family.
- Equivariant Jiang–Su stability —
- The property that a group action absorbs the trivial action on up to cocycle conjugacy.
- Strict comparison —
- The comparison condition that the manuscripts use as a sufficient condition for Jiang–Su absorption under specified hypotheses.
- Nuclear dimension —
- The dimension quantity whose finiteness is related here to Jiang–Su stability and for which the family states a bound after tensoring with .
- Uniform property Γ —
- The tracial regularity property obtained in the Lean-marked companion from the stated real-rank-zero assumption.
Keywords : Jiang–Su stability · Cuntz comparison · Nuclear dimension
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case
05/10/2026Every 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.
Cuntz comparison and Jiang–Su absorption
23/09/2026We 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.
Nuclear dimension and Jiang–Su stability without elementary subquotients
23/09/2026For separable nuclear C∗-algebras with no nonzero elementary ideal subquotients, finite nuclear dimension is equivalent to Jiang–Su stability. More generally, for every separable nuclear A0. This proves Robert and Tikuisis's Conjecture (C1) and the nuclear-dimension equivalence in the nonsimple Toms–Winter regularity question.
Tracial projection methods and uniform property Gamma
23/09/2026For 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.