Reduced $C^*$-Algebras of Free Orthogonal Quantum Groups

Definitions and mathematical statement

The supplied record uses $\mathrm{FO}_n$ in its cleaned statement and $F O_n^+$ in its original statement, without defining either notation. The following formulation adopts the conventional free orthogonal quantum group interpretation suggested by the title; this interpretation requires human verification against the cited report.

For each positive integer $n$, let $A_n=C^u(O_n^+)$ be the universal unital complex $C^*$-algebra generated by elements $u_{ij}=u_{ij}^*$, $1\leq i,j\leq n$, satisfying \[ \sum_{k=1}^n u_{ik}u_{jk}=\delta_{ij}1, \qquad \sum_{k=1}^n u_{ki}u_{kj}=\delta_{ij}1 \quad (1\leq i,j\leq n). \] Equip this algebra with the coproduct \[ \Delta_n(u_{ij})=\sum_{k=1}^n u_{ik}\otimes u_{kj}. \] Let $h_n$ be its Haar state, characterized by \[ (h_n\otimes\mathrm{id})\Delta_n(a) =(\mathrm{id}\otimes h_n)\Delta_n(a) =h_n(a)1 \quad(a\in A_n), \] and let $\pi_{h_n}$ be its GNS representation. Define the reduced algebra \[ B_n=\overline{\pi_{h_n}(A_n)}^{\lVert\cdot\rVert}. \] Under the stated interpretation, $\mathrm{FO}_n$ denotes the discrete quantum group dual to $O_n^+$, and the source expression $C_r^*(\mathrm{FO}_n)$ denotes $B_n$.

Question. Given $M,N\in{\mathbb{Z}}_{\geq 1}$ with $M\neq N$, determine whether there exists a complex $C^*$-algebra isomorphism \[ \Phi:B_M\longrightarrow B_N. \] Here an isomorphism is a bijective complex-linear map preserving multiplication and the involution; no compatibility with the coproduct is imposed. The question concerns the underlying reduced $C^*$-algebras. All positive ranks, including $1$ and $2$, are retained as supplied. The record does not clarify whether an additional rank restriction was intended or reconcile its two quantum group notations.

Short English statement

For distinct positive integers $M$ and $N$, determine whether the reduced $C^*$-algebras associated with the free orthogonal quantum groups of those ranks are isomorphic. This formulation uses an explicitly stated interpretation of the source notation. The supplied open label is dated curation, not an independently verified assertion that every such pair remains unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004259$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2019.
• Supplied report citation: Mini-Workshop: Operator Algebraic Quantum Groups (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/45, pp. 2821--2867. https://doi.org/10.4171/owr/2019/45. The displayed year 2020 and the 2019 provenance are retained as supplied.
• Supplied literature source labelled "OWR 2019 45": https://ems.press/content/serial-article-files/46823.
• Supplied literature source labelled "Voigt": https://arxiv.org/abs/0911.2999. No further bibliographic details are supplied.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified here. The supplied status is "open". The assessment states that $K$-theory computations for reduced free-group $C^*$-algebras do not recover the rank parameter and that the report leaves the distinct-rank isomorphism question open. Its reference to free-group algebras is retained as a source qualification; its applicability to the free orthogonal quantum group question requires review.
• The record labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is a supplied verification claim, not a verification performed for this draft. The background expressly requests independent verification of time-sensitive status and cited claims.