Result 207, Algebra

The ℓ¹-Bass conjecture for all discrete groups

Proves the ℓ¹-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $\ell^1(G)$ are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.

Lean formalization Proof

The bigger picture

Why it matters

Matrices that remain unchanged when squared encode important algebraic structures. The manuscript claims that, for every discrete group, a trace of such matrices can detect only finitely many classes of elements with finite order.

What changes?

The manuscript reports this restriction for idempotent matrices of every finite size over the complex l1 group algebra, whose elements assign absolutely summable complex coefficients to group elements. Idempotent means that squaring leaves the matrix unchanged. The Hattori-Stallings trace sums diagonal coefficients by conjugacy class, collecting elements related by conjugation. The claim is that only finitely many classes contribute, and each consists of finite-order elements: elements with some positive power equal to the identity. No restriction on the discrete group is imposed.

What does that help mathematicians do?

For torsion-free groups, which have no nonidentity elements of finite order, the trace can therefore be supported only at the identity. The companion manuscript also claims that such groups have no group-ring idempotents except 0 and 1 over any commutative domain with identity and characteristic zero. This rules out nontrivial projection elements in those rings. It does not say that all idempotent matrices are trivial, an important distinction for studying modules.

Are there practical applications?

The immediate value is foundational: the trace restriction constrains invariants of finitely generated projective modules, algebraic objects represented by idempotent matrices. Researchers could use it to rule out proposed trace patterns involving infinite-order elements. The supplied abstracts describe structural consequences for group algebras, not computational speedups or practical deployments.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The ℓ¹-Bass Conjecture for Discrete Groups

October 5, 2026 33 pages

We prove the ℓ¹-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ¹ group algebra is supported on finitely many conjugacy classes of finite-order elements.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026,
  author = {{OpenAI}},
  title = {{The $\ell^1$-Bass Conjecture for Discrete Groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf}{OAI:The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026}},
  year = {2026}
}

The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture

September 24, 2026 33 pages Main result formalized in Lean

We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in $RG$ are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026,
  author = {{OpenAI}},
  title = {{The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf}{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/207.md.

The ℓ¹-Bass conjecture for all discrete groups

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture

Scope

The formalization proves the complex Bass trace conjecture for every group: the Hattori–Stallings trace of each virtual class of finitely generated projective right $\mathbb C[G]$ -modules vanishes on conjugacy classes of infinite-order elements. No finiteness, countability, or geometric hypothesis on $G$ is imposed.

For torsion-free $G$ , it identifies the trace on $K_0(\mathbb C[G])$ with the integral augmentation rank and proves the characteristic-zero Kaplansky idempotent conjecture: for every commutative unital domain $R$ of characteristic zero, an idempotent in $R[G]$ is $0$ or $1$ .

Comparator links

Result	Comparator statement
Complex Bass trace vanishing and support	BassTrace.lean
Trace rank and characteristic-zero Kaplansky idempotents for torsion-free groups	BassTorsionFree.lean