Stable Rank One for Simple Reduced Group $C^*$-Algebras

Definitions and mathematical statement

Let $G$ be a discrete group, with no countability assumption. Let $\ell^2(G)$ be the Hilbert space of square-summable complex functions on $G$. The left regular representation is given by \[ (\lambda(g)\xi)(h)=\xi(g^{-1}h) \qquad(g,h\in G,\ \xi\in\ell^2(G)). \] The reduced group $C^*$-algebra is the unital complex $C^*$-algebra \[ A_G=C^*_{\lambda}(G) :=\overline{\operatorname{span}_{{\mathbb{C}}}\{\lambda(g):g\in G\}}^{\|\cdot\|} \subseteq B(\ell^2(G)), \] where the closure is taken in operator norm. Its unit is $\lambda(e)$, where $e$ is the identity of $G$. A nonzero $C^*$-algebra is simple if its only norm-closed two-sided ideals are $\{0\}$ and the whole algebra. For a unital $C^*$-algebra $A$, write \[ A^{\times}=\{b\in A:\text{there exists }c\in A\text{ with }bc=cb=1_A\}. \] Here stable rank one means topological stable rank one, equivalently that $A^{\times}$ is norm dense in $A$.

Question. Is it true that, for every discrete group $G$, \[ C^*_{\lambda}(G)\text{ simple} \quad\Longrightarrow\quad \overline{\left(C^*_{\lambda}(G)\right)^{\times}}^{\|\cdot\|} =C^*_{\lambda}(G)? \] Equivalently, whenever $C^*_{\lambda}(G)$ is simple, must every $a\in C^*_{\lambda}(G)$ and every real $\varepsilon>0$ admit an invertible $b\in C^*_{\lambda}(G)$ such that $\|a-b\|<\varepsilon$?

The supplied record labels this question open. Its dated literature assessment states that stable rank one is known for important families, including free products and many acylindrically hyperbolic groups, while the general question remains open. The precise hypotheses for those background results are not supplied; this draft does not interpret them as assertions about all groups in either family. The status and background assessment are supplied curation, not independently verified conclusions.

Short English statement

Does every discrete group whose reduced group $C^*$-algebra is simple have the property that every element of that algebra can be approximated arbitrarily closely in norm by invertible elements? No countability assumption on the group is imposed. The supplied open status requires independent review.

Sources

• Local problem registry, supplied record for permanent UnsolvedMath ID $30004210$, source problem code catalogue entry, titled "Stable Rank One for Simple Reduced Group $C^*$-Algebras," in the Oberwolfach Reports Open Problems collection. The project registry paths are data/unsolvedmath/problems.json and catalog/problems.json. No individual proposer is identified in the supplied record; its proposed year is 2019.
• Supplied report citation: C*-Algebras (2020), Oberwolfach Reports, pp. 2257--2332, DOI 10.4171/owr/2019/37. https://doi.org/10.4171/owr/2019/37. The literature-source label is "Oberwolfach Report 37, stable rank." The supplied title year 2020 and the year 2019 appearing in the DOI and provenance require bibliographic clarification; they are retained as supplied.
• Supplied literature reference: "Gerasimova and Osin, invertible elements," DOI 10.1016/j.jfa.2020.108689. https://doi.org/10.1016/j.jfa.2020.108689. No full title or theorem locator is supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and reports stable rank one for important families, including free products and many acylindrically hyperbolic groups. This is supplied dated curation derived from the Oberwolfach Reports record, not independently verified source consultation or a current literature assessment. The supplied record requests independent verification of time-sensitive status and cited claims.