Non-MF groups and non-finite full group C*-algebras

Let $\Gamma$ be a property (T) group admitting an injective, non-surjective endomorphism and let $G$ be the associated ascending HNN-extension. Let $W=\left(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\right)\rtimes G.$ We show that $W$ is not an MF group and that $C^*(G)$ is not a finite $C^*$ -algebra. The ideas and proofs were generated by ChatGPT 5.6 Sol, we have only refined their arguments in a hopefully more palatable form.

Result: Disproved(see note)
Status: Resolved
AI contribution: AI-discovered
Method: Construction
Field: Operator algebra
Posed by: Whether a stably finite C*-algebra must be MF, and whether every full group C*-algebra is finite; questions of the C*-algebra literature the paper's introduction cites
Year posed: —
Years open: —
Solved: 2026-08-28
Model: GPT-5.6 Sol
Vendor: OpenAI
Collaborators: Caleb Eckhardt
Verification: Unreviewed
Publication: Preprint
Significance: 25 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

If $\Gamma$ is a non-coHopfian property-(T) group and $G$ is its ascending HNN extension, then the full group algebra $C^*(G)$ is not finite. Indeed, Kazhdan projections $p_H<p_\Gamma$ become unitarily equivalent under the stable letter, which cannot occur inside a finite $C^*$ -algebra.

For $W=\left(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\right)\rtimes G,$ the paper proves more strongly that every homomorphism $W\to U\!\left(\prod M_{d_n}/\bigoplus M_{d_n}\right)$ kills an explicit nonidentity element $b_\gamma$ . Hence $W$ is not MF. Consequently $C_r^*(W)$ is an explicit stably finite but non-MF $C^*$ -algebra.

What the AI did

Caleb Eckhardt states that the mathematical ideas and proofs were generated by ChatGPT 5.6 Sol. He prompted Sol to seek constructions starting from the non-sofic examples of Kun and Thom, themselves building on recent OpenAI examples. Sol developed the core arguments, including the property-(T)/Kazhdan-projection mechanism and a workaround using a rescaled Hilbert-Schmidt representation and a nontrivial $1$ -cocycle to prove the generalized wreath-product group is non-MF. Eckhardt internalized, refined, and rewrote the arguments.

Verification

Unreviewed. arXiv 2608.28772 (seven pages) read here: "The ideas and proofs were generated by ChatGPT 5.6 Sol, we have only refined their arguments"; Eckhardt internalised and rewrote them and takes responsibility. Willett, Fournier-Facio, Dogon and Shulman are thanked for input and one consequence, which is comment rather than a check of the complete proof; not refereed; Sauer's Lean work on related examples is separate.

Sources

PaperarXivarXiv 2608.28772

Submitted by VibeGene on 06 Sep 2026

Non-MF groups and non-finite full group C*-algebras

What was actually shown

What the AI did

Verification

Sources

Discussion