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The trace cone classifies Razak–Jacelon stabilizations
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Abstract
We prove that the canonical topological cone of all extended lower-semicontinuous tracial weights determines a separable nuclear C∗-algebra after tensoring with the Razak–Jacelon algebra and the compact operators, answering Robert's trace-cone classification question positively. The isomorphism realizes the prescribed cone map, with arbitrary ideal structure and without a density assumption on the finite domains of the weights.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 301 (release, 2026-10-06). The release's Lean development covers only the ideal-transport step: a trace-cone isomorphism induces an order isomorphism of closed ideals (with each trace's finite and zero ideals), and the zero/infinity ideal traces are exactly the additive idempotents; spatial classification, existence, uniqueness and intertwining are not formalized. [family-level]
- OpenAI math release, TraceIdealTransportSupport.lean (release, 2026-10-06). The comparator statement for the ideal-transport result described in the scope notes.
Related prior literature mentioned by others
- Luis Santiago, Reduction of the dimension of nuclear C*-algebras (arXiv:1211.7159, 30 November 2012). Linked as background by the AI Proved What? episode; it states Robert's question as Conjecture 1.1 (T(A) ≅ T(B) iff A⊗W⊗K ≅ B⊗W⊗K), noting it was known for AF and O_2-absorbing algebras, and proves stable rank one after tensoring with the Razak–Jacelon algebra for a class containing limits of separable type I algebras.
- Christopher Schafhauser, Aaron Tikuisis and Stuart White, Nuclear C*-algebras: 99 problems (arXiv:2506.10902, 2025; revised 8 May 2026). The AI Proved What? episode identifies Problem LXVIII in this survey as the trace-cone classification question addressed by the manuscript.
Possibly relevant literature (found by the agent, not checked)
- George A. Elliott and Jeffrey Im, Colored isomorphism of classifiable C*-algebras (Canadian Journal of Mathematics 76, 216–245, 2024; published online 12 December 2022). Remark 4.2 recovers ideal lattices from the additive idempotents of the lower-semicontinuous trace cone, a comparison for the manuscript's ideal-recovery step.
- Laurent Cantier, Tracially reflexive C*-algebras (arXiv:2605.22683, May 2026). States Robert's conjecture for the scaled cone up to W-tensorisation, says it should otherwise be stated up to stable isomorphism, and notes that the idempotents of T(A) encode the ideal lattice; this bears on the paper's stabilization by K and its ideal-weight encoding.
- Leonel Robert, Classification of inductive limits of 1-dimensional NCCW complexes (arXiv:1007.1964, 2010–2012). Santiago's paper reduces the conjecture for limits of homogeneous algebras to this classification, giving an earlier non-simple case against which Theorem 1.1 can be compared.
- Mark Moodie and Leonel Robert, Cones of traces arising from AF C*-algebras (arXiv:2009.09033, 2020). Characterizes the non-cancellative cones arising as extended trace cones of AF algebras via a duality with Cuntz semigroups with real multiplication; relevant to the paper's use of the full extended-trace cone, including ideal weights.
Digests and expositions
- AI Proved What?, The cone decides? Robert’s trace-cone question (episode 301, October 2026). This AI-assisted video and transcript illustrate extended traces and ideal recovery, outline the claimed existence-and-uniqueness argument, and distinguish the supporting Lean development from the classification theorem. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 301
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026,
author = {{OpenAI}},
title = {{The trace cone classifies Razak--Jacelon stabilizations}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026.pdf}{OAI:The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/301.md. The Lean statements have not been checked by people against the paper's claims.
Family 301: Trace cones and Razak–Jacelon stabilization
Classifies separable nuclear complex C∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.
This is the only paper in this family.