The ℓ¹ Bass conjecture

About 16 years old · traced to 2010

Let $G$ be a discrete group, let $[G]$ be its set of conjugacy classes, and let $\mathrm{FC}(G)$ be the subset consisting of conjugacy classes represented by elements of finite order. The $\ell^1$ Hattori–Stallings trace is

HS^1:K_0(\ell^1(G))\longrightarrow\ell^1([G]).

The $\ell^1$ Bass conjecture. The image of $HS^1$ is contained in

\bigoplus_{\mathrm{FC}(G)}C{C},

the subspace of functions finitely supported on conjugacy classes of elements of finite order. The paper proves that the Bost conjecture implies this statement, while its general validity is not resolved in the supplied text.

References

Primary source

A. J. Berrick and I. Chatterji And G. Mislin, “From acyclic groups to the bass conjecture for amenable groups”, arXiv:1004.1941 (2010).

Posted 1 month ago

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

RemarkAI-assistedClaimed by OpenAI. For every discrete group, the manuscript claims that the Hattori–Stallings trace of every idempotent matrix over its complex l1 group algebra is supported on finitely many conjugacy classes of finite-order elements. By additivity this gives the page’s stated K0 image containment. This is the l1 Bass claim; it does not assert the stronger integral trace is supported only on the identity class.See full solutionHide full solution

Claimed by OpenAI. For every discrete group, the manuscript claims that the Hattori–Stallings trace of every idempotent matrix over its complex l1 group algebra is supported on finitely many conjugacy classes of finite-order elements. By additivity this gives the page’s stated K0 image containment. This is the l1 Bass claim; it does not assert the stronger integral trace is supported only on the identity class.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf

OpenAI-207-01-The-Bass-Conjecture-for-Discrete-Groups.pdf502,912 bytesOpen