Non-MF groups and non-finite full group C*-algebras
resolvedconfidence 70%
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
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.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-5.6 Sol
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.
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.
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.
Known method families
construction (source-reported)
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
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AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.