Operator Models for Universal Hopf $C^*$-Algebras

Definitions and mathematical statement

The source requests concrete operator models and spectral-measure faithfulness criteria for universal Hopf $C^*$-algebras beyond the basic $A_u(n)$, $A_o(n)$ and $A_s(n)$ examples, especially quantum automorphism algebras of finite graphs. The original refers to Theorem 2, whose precise statement is absent.

In the standard compact-matrix-quantum-group convention, $A_u(n)$ is generated by a matrix $u$ with $u$ and its entrywise conjugate unitary; $A_o(n)$ has a self-adjoint-entry orthogonal generating matrix; and $A_s(n)$ has a magic unitary, meaning projections with each row and column summing to one. Their coproduct satisfies $\Delta(u_{ij})=\sum_k u_{ik}\otimes u_{kj}$. For a finite graph $X$ with $n$ vertices and adjacency matrix $D_X$, the usual quantum automorphism algebra is \[ A(X)=A_s(n)/\langle uD_X-D_Xu\rangle. \] Graph and universal-completion conventions require source confirmation.

A concrete operator model is a unital $*$-homomorphism $\pi:A(X)\to B(\mathcal H)$, given by operators satisfying the defining relations. For this magic-unitary case the main character $\chi=\sum_i u_{ii}$ is self-adjoint. Given a specified state $\varphi$ on the model algebra, its spectral probability measure is determined by \[ \int t^r\,d\mu_{\pi,\varphi}(t)=\varphi(\pi(\chi)^r) \quad(r\geq0). \] The Haar state $h$ similarly gives $\mu_h$ for $\chi$. The question asks for an analogue of the source's criterion relating such spectral data to the appropriate faithfulness property, together with usable models for the target algebras. It does not supply the needed state, criterion, moment conditions or quantifier over graph classes.

Faithfulness of $\pi$ as an injective $C^*$-representation must be distinguished from inner faithfulness, which excludes a nontrivial Hopf quotient through which the model factors. Universal and reduced completions and faithfulness of the Haar state are also different issues. No equality-of-measures iff injectivity claim is asserted without Theorem 2. For non-self-adjoint characters in other quantum groups, even the spectral-data convention may need $*$-moments or another specified observable.

The supplied literature triage checked 21 August 2026 says the basic examples do not resolve the requested general analogue. It supplies no general theorem or authoritative later resolution, and this dated finding is not an independent current literature assessment.

Short English statement

Find operator realizations and a suitable spectral-data criterion for quantum groups beyond the basic universal examples, especially finite-graph quantum automorphisms. The omitted Theorem 2 is needed to specify the state and meaning of faithfulness; injective representations and inner-faithful models are distinct.

Sources

• Local UnsolvedMath registry, permanent ID 30000309, exact source code catalogue entry; Operator Models for Universal Hopf $C^*$-Algebras. Proposed year 2005, status open. The corrected_verified statement restores the model/faithfulness target and graph class; its claimed inspection is supplied metadata.
• Supplied report citation: $C^*$-Algebren (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/41, pp. 2305--2374, https://doi.org/10.4171/owr/2005/41. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21, citing only "original Oberwolfach report", https://doi.org/10.4171/owr/2005/41. It reports that the basic examples do not settle the general question. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no full Theorem 2 or separate later reference is supplied.