Selflessness of the Reduced $C^*$-Algebra of $\mathrm{SL}_3(\mathbb{Z})$

Definitions and mathematical statement

Let \[ \Gamma=\mathrm{SL}_3({\mathbb{Z}})=\{g\in M_3({\mathbb{Z}}):\det(g)=1\}. \] Let $\ell^2(\Gamma)$ be the Hilbert space of square-summable complex-valued functions on $\Gamma$. The left regular representation is defined by \[ (\lambda(g)\xi)(h)=\xi(g^{-1}h) \qquad(g,h\in\Gamma,\ \xi\in\ell^2(\Gamma)). \] The reduced group $C^*$-algebra is \[ A=C_r^*(\Gamma) =\overline{\operatorname{span}_{{\mathbb{C}}}\{\lambda(g):g\in\Gamma\}}^{\|\cdot\|} \subseteq B(\ell^2(\Gamma)). \] The supplied question is:

Is $C_r^*(\mathrm{SL}_3({\mathbb{Z}}))$ selfless, in the sense intended by the cited Oberwolfach report?

Unresolved definition. The supplied record does not give the definition of selflessness. Its extracted text includes the relation $\Psi\circ\iota_A=\Delta$, but omits the domains, codomains, quantifiers, and surrounding construction. Consequently, this draft cannot give a self-contained quantified definition of the central property without additional source evidence. In particular, no choices concerning states, free products, ultrapowers, or embeddings are imposed here.

Supplied context and qualifications. The record states that reduced group $C^*$-algebras of cocompact lattices in $\mathrm{SL}_3({\mathbb{R}})$ are selfless. Its supplied literature assessment distinguishes $\mathrm{SL}_3({\mathbb{Z}})$ as a nonuniform higher-rank lattice not covered by that result. These are supplied contextual claims, not independently verified results in this draft.

The extracted discussion describes infinite free products, including $C_r^*(F_\infty)$, as examples of selfless algebras, where $F_\infty$ is the free group on countably infinitely many generators. It attributes to Amrutam--Gao--Kunnawalkam Elayavalli--Patchell a proof of selflessness, and hence strict comparison, for large classes of reduced group algebras, described in the excerpt as including all hyperbolic groups and many acylindrically hyperbolic groups. It also says their work settled strict comparison for $C_r^*(F_2)$, where $F_2$ is the free group on two generators, and motivated seeking examples among higher-rank lattices. The excerpt supplies neither the full hypotheses of these claims nor complete references for its numbered citations, so their scope requires source review.

The original extraction additionally contains an incomplete sentence asserting that $(\mathrm{RD})$ remains open for many nonuniform lattices. Its connection to the displayed question is unclear; no $(\mathrm{RD})$ hypothesis or additional question is inferred from this fragment.

Short English statement

The question asks whether the reduced group $C^*$-algebra of the integer determinant-one $3\times3$ matrices is selfless. The supplied record labels the question open and distinguishes it from a claimed result for cocompact lattices in $\mathrm{SL}_3({\mathbb{R}})$. The definition of selflessness is missing from the supplied evidence, so the intended property requires source review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006424$, exact source problem code catalogue entry, title "Selflessness of the Reduced $C^*$-Algebra of $\mathrm{SL}_3({\mathbb{Z}})$." The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2025.
• Supplied report citation: C*-Algebras (2026), Oberwolfach Reports, pp. 1887--1948, DOI 10.4171/owr/2025/35. Source URL: https://doi.org/10.4171/owr/2025/35. This is also the supplied literature source labeled "Oberwolfach Report 35." The report title's year and the DOI's year differ in the supplied citation and have not been reconciled.
• Supplied literature source labeled "Strict comparison primary paper": https://doi.org/10.1007/s00222-025-01366-5. The extracted discussion attributes the relevant breakthrough to Amrutam--Gao--Kunnawalkam Elayavalli--Patchell; a complete bibliographic entry and a verified correspondence with its citation [1] are not supplied.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It labels the problem open, states that selflessness is proved for cocompact lattices in $\mathrm{SL}_3({\mathbb{R}})$, and says the nonuniform lattice $\mathrm{SL}_3({\mathbb{Z}})$ is not covered and the indicated higher-rank lattice direction remains open. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The extracted discussion refers to [1] and [5], but their complete reference entries are absent from the supplied record. The example involving $C_r^*(F_\infty)$ is attached to [5].