Construction of groups that are not MF
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Which finitely generated groups fail to be MF, meaning their finite matrix models cannot provide the required norm approximation? The recorded route starts from a property-(T) group with an injective nonsurjective endomorphism and passes to an ascending HNN extension and a generalized wreath product.
Why it matters and what counts as progress
Why it matters: An explicit non-MF group makes an operator-algebraic approximation obstruction concrete rather than relying only on abstract existence.
What counts as progress: Give an explicit non-MF group and a separating element annihilated by every proposed matrix-ultraproduct representation.
How the problem developed
- 1997 · Foundations [1]
Bruce Blackadar and Eberhard Kirchberg Introduced MF C*-algebras, defined through embeddings into matrix ultraproducts; the group approximation question asks for an analogous representation property.
- 2017 · Partial result [1]
Aaron Tikuisis, Stuart White and Wilhelm Winter Proved amenable groups satisfy the stronger C*-MF approximation property.
- 28 Aug 2026 · Reported claim [1]
Caleb Eckhardt and ChatGPT 5.6 Sol Constructed explicit groups that are not MF and whose full group C*-algebras are not finite; the paper says the AI generated the ideas and proofs, which Eckhardt refined and reviewed.
AI contribution
Non-MF groups and non-finite full group C*-algebras
From a property-(T) group Γ with an injective non-surjective endomorphism, Eckhardt forms its ascending HNN extension G and generalized wreath product W=(⊕_(G/Γ) Z/2Z)⋊G. His Theorem 4.3 gives a specified nonidentity element killed by every homomorphism of W to a matrix-ultraproduct quotient, showing W is not MF.
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