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Robert-Tikuisis Conjecture (C1): finite nuclear dimension implies Jiang-Su stability without elementary subquotients

For simple algebras, finite nuclear dimension and Jiang-Su stability are equivalent away from the elementary case. Robert and Tikuisis studied the nonsimple setting, where the natural exclusion is that no ideal subquotient I/JI/J is a nonzero elementary algebra K(H)\mathcal K(H) (nowhere scattered algebras). Their Conjecture (C1) predicts that a separable nuclear C*-algebra of finite nuclear dimension with no nonzero elementary ideal subquotients absorbs the Jiang-Su algebra Z\mathcal Z. Does finite nuclear dimension imply A≅A⊗ZA\cong A\otimes\mathcal Z for every separable nuclear C*-algebra with no elementary subquotients?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Operator algebras; nuclear dimension of nonsimple C*-algebras
Posed by
Leonel Robert and Aaron Tikuisis
Year posed
2017
Years open
9y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for separable nuclear AA with no nonzero elementary ideal subquotients, dim⁡nucA<∞\dim_{nuc}A<\infty, A≅A⊗ZA\cong A\otimes\mathcal Z and dim⁡nucA≤1\dim_{nuc}A\le1 are equivalent. Theorem 1.3: dim⁡nuc(A0⊗Z)≤1\dim_{nuc}(A_0\otimes\mathcal Z)\le1 for every separable nuclear A0A_0. Theorem 1.4 answers Kirchberg-Rordam's Question 6.2 (full orthogonal positive elements in a finite maximal tensor power of a unital algebra without characters). It does not address the strict-comparison part of the nonsimple problem, which the Cuntz-comparison companion treats.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry's principal is 'Nuclear dimension and Jiang-Su stability without elementary subquotients' (September 23, 2026), one of four manuscripts in the family.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Conjecture (C1). For separable nuclear algebras with no nonzero elementary ideal subquotients it proves finite nuclear dimension, Z\mathcal Z-stability and nuclear dimension at most 1 equivalent; the forward implication is (C1). The proof was not refereed and is not formalised.

Sources

Changelog1 change

Discussion