Single Generation and Free-Group $C^*$-Superrigidity

Definitions and mathematical statement

A complex $C^*$-algebra $A$ is simple if it is nonzero and its only closed two-sided ideals are $\{0\}$ and $A$; it is separable if it has a countable norm-dense subset. For $a\in A$, let $C^*(a)$ denote the smallest norm-closed $*$-subalgebra of $A$ containing $a$. Say that $A$ is singly generated if $A=C^*(a)$ for some $a\in A$; the generator need not be self-adjoint.

The first question is whether \[ \forall A\;\left[A\text{ is a unital simple separable complex }C^*\text{-algebra}\ \Longrightarrow\ \exists a\in A\; A=C^*(a)\right]. \] The supplied dated curation reports a counterexample to this assertion; that report is not independently verified here.

Let $F_2=\langle s,t\mid\ \rangle$ be the free group on two generators. For a discrete group $\Gamma$, its left regular representation on $\ell^2(\Gamma)$ is defined by $\lambda_\Gamma(g)\delta_h=\delta_{gh}$, and its reduced group $C^*$-algebra is \[ C_r^*(\Gamma)=\overline{\operatorname{span}\{\lambda_\Gamma(g):g\in\Gamma\}}^{\|\cdot\|}\subseteq B(\ell^2(\Gamma)). \] Using the standard meaning of a hyperbolic group, namely a finitely generated group whose Cayley graph for a finite generating set is Gromov hyperbolic, the group-determination question is whether \[ \forall\Gamma\;\left[\Gamma\text{ is a hyperbolic group and }C_r^*(\Gamma)\cong C_r^*(F_2)\text{ as }C^*\text{-algebras}\ \Longrightarrow\ \Gamma\cong F_2\text{ as groups}\right]. \] No torsion-free hypothesis is supplied. The source also asks about a possibly stronger $C_r^*$-superrigidity property, but does not define that property or specify restrictions on the algebra isomorphism. Consequently, the displayed implication formalizes only the group-determination question; the additional superrigidity target remains unresolved at the level of formulation.

Short English statement

The questions ask whether every unital simple separable $C^*$-algebra has one generator, and whether a hyperbolic group with reduced group $C^*$-algebra isomorphic to that of $F_2$ must itself be isomorphic to $F_2$. The supplied curation reports a negative answer to the first question and describes the second as open. The stronger intended meaning of superrigidity is unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30006428$, source code catalogue entry, title: Single Generation and Free-Group $C^*$-Superrigidity. The supplied record lists proposed year 2025 and identifies no individual proposer.
• Supplied report citation: C*-Algebras (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/35, pp. 1887--1948. https://doi.org/10.4171/owr/2025/35. This is the supplied workshop-report provenance; the year discrepancy between the title, DOI, and proposed year is unresolved.
• Supplied literature source, labeled Elliott--Li--Niu counterexample: https://arxiv.org/abs/2608.13707. The supplied assessment attributes to an August 2026 construction a simple unital separable AH (approximately homogeneous) algebra that is not singly generated.
• Supplied literature source, labeled Oberwolfach Report 35: https://doi.org/10.4171/owr/2025/35.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It labels the combined record partially solved, reports the universal single-generation assertion disproved, and describes the separate $C^*$-superrigidity question for $F_2$ within hyperbolic groups as open. The supplied background explicitly requests independent verification of these time-sensitive statuses and cited claims.