AF Full Group $C^*$-Algebras and Local Finiteness

Definitions and mathematical statement

Let $G$ be a countable discrete group, allowing finite groups. Its full group $C^*$-algebra $C^*(G)$ is the completion of the complex group algebra ${\mathbb{C}}[G]$ in the maximal norm \[ \|a\|_{\max}=\sup_{\pi}\|\pi(a)\|, \] where $\pi$ ranges over all unitary representations of $G$ on complex Hilbert spaces, extended linearly to ${\mathbb{C}}[G]$. The word "full" follows the supplied problem title.

A $C^*$-algebra $A$ is approximately finite-dimensional (AF) if there is an increasing sequence of finite-dimensional $C^*$-subalgebras $A_n\subseteq A$ such that \[ A=\overline{\bigcup_{n=1}^{\infty}A_n}^{\|\cdot\|}. \] A group $G$ is locally finite if every finitely generated subgroup is finite; equivalently, \[ \forall F\subseteq G\quad \left(F\text{ finite}\Longrightarrow |\langle F\rangle|<\infty\right). \]

The question is whether the following implication holds for every countable discrete group $G$: \[ C^*(G)\text{ is AF} \quad\Longrightarrow\quad G\text{ is locally finite}. \] The supplied record describes this as the converse of the known implication from local finiteness to the AF property. Its dated literature assessment reports that the converse is known for nilpotent groups and for finitely generated elementary amenable groups, while the general countable discrete case remains open. These are supplied source and status claims, not independently verified conclusions of this draft. The report-year discrepancy and the absence of precise references for the stated special cases remain unresolved.

Short English statement

Must every countable discrete group whose full group $C^*$-algebra is approximately finite-dimensional have only finite finitely generated subgroups? The supplied curation labels the problem partially solved, reporting affirmative results for nilpotent groups and finitely generated elementary amenable groups, with the general case remaining open.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003206$, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The supplied record identifies the proposed year as 2016 and supplies no individual proposer.
• Supplied source citation, with mathematical typography normalized: $C^*$-Algebras (2017), Oberwolfach Reports, pp. 2269--2345, DOI 10.4171/owr/2016/40. https://doi.org/10.4171/owr/2016/40. The citation's 2017 label differs from the supplied proposed year and the year appearing in the DOI; this draft does not resolve that discrepancy.
• Supplied literature source: "OWR Operator Algebras report." https://ems.press/content/serial-article-files/46645?nt=1.
• Supplied literature source: "Nuclear C star algebras survey." https://ems.press/journals/emss/articles/14298154.
• Literature assessment dated exactly 2026-08-21: the supplied curation reports that the converse is known for nilpotent groups and finitely generated elementary amenable groups, but remains open for arbitrary countable discrete groups. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified and says that the official report was checked and describes the converse as unknown in general. This records the supplied verification claim; no source visit or independent verification is asserted here. The supplied problem status is partially_solved.