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Math-2821 · mathematics · Analysis

Kadison–Kaplansky projection conjecture

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every torsion-free discrete group G, are 0 and 1 the only projections in the scalar reduced group C*-algebra C*r(G)? This concerns the completed complex C*-algebra, rather than the algebraic group ring or only matrix projections.

Short English statement

For every torsion-free discrete group G, are 0 and 1 the only projections in the scalar reduced group C*-algebra C*r(G)? This concerns the completed complex C*-algebra, rather than the algebraic group ring or only matrix projections.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2821-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2821

Manuscript 1 (family 285, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed counterexample to that assertion in full.

Kadison–Kaplansky projection conjecture

OpenAI's claim

OpenAI claims the existence of a nontrivial scalar projection inside the reduced C*-algebra of a finitely generated torsion-free group, contradicting the Kadison--Kaplansky assertion. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims a discrete group, torsion-free and finitely generated, $G$ whose reduced group $C^*$-algebra contains a scalar projection $e=e^*=e^2$ with \[ 0<\tau_G(e)<\tfrac12,\qquad\tau_G(x)=\langle x\delta_1,\delta_1\rangle. \] Thus $e$ is neither $0$ nor $1$. This concerns a scalar projection in the operator-norm completion of the left regular group algebra, and supplies a counterexample to the torsion-free projection assertion. The manuscript also uses the integral trace restriction on reduced assembly for torsion-free groups to place this projection class outside the assembly image.

Outline of the argument

Manuscript 1 (family 285, 2026-09-23): A magnetic tower model produces a separated spectral band carrying a matrix projection of positive trace below one half. Graph sampling and small-cancellation recovery preserve this band after the model is represented on the group, with intersection losses bounded by trace estimates. A free-product compression converts the matrix projection into a scalar one while retaining torsion-freeness.

Formal verification

Family 285 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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