Twisted Groupoid Reconstruction of $C^*$-Algebra Pairs

Definitions and mathematical statement

A pair of $C^*$-algebras means an ordered pair $(A,B)$; any inclusion or additional structure required of this pair must be specified by the reconstruction theorem referenced below. For a groupoid $G$ with unit space $G^{(0)}$, source map $s$, and range map $r$, the isotropy group at $x\in G^{(0)}$ is \[ G_x^x=\{g\in G:s(g)=r(g)=x\}. \] The supplied passage refers to a reconstruction result called "Theorem 1," but does not include its statement. Let $\mathcal G_1$ denote the class of groupoids specified in that theorem, and let $\mathcal P_1$ denote the collection of $C^*$-algebra pairs appearing there. These are source-dependent placeholders, not independently specified classes.

The source asks for the following three related developments:

1. Characterize $\mathcal P_1$ by intrinsic $C^*$-algebraic properties. The requested characterization has the schematic form \[ (A,B)\in\mathcal P_1 \quad\Longleftrightarrow\quad (A,B)\text{ satisfies the proposed intrinsic conditions}, \] with the admissible pairs and their additional structure determined by Theorem 1.
2. Obtain a version of Theorem 1 for twisted $C^*$-algebras over the same class $\mathcal G_1$ of groupoids.
3. Determine whether such a twisted version can be used to weaken the hypotheses on the isotropy groups in Theorem 1.

The supplied summary describes the setting as quasi-Cartan groupoid reconstruction. However, neither the meaning of quasi-Cartan in this setting, the reconstruction construction and conclusion, the class $\mathcal G_1$, the permitted twists and associated $C^*$-algebra completion, nor the existing isotropy hypotheses are supplied. Consequently, this is a source-dependent statement of the research targets; a fully specified quantified formulation requires the missing theorem and its definitions.

Short English statement

Characterize the $C^*$-algebra pairs in the referenced quasi-Cartan reconstruction theorem, extend that theorem to twisted groupoid $C^*$-algebras over its original groupoid class, and determine whether the extension permits weaker isotropy assumptions. Theorem 1 is absent from the supplied record, so its exact hypotheses and conclusions remain unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $30005194$, source problem code catalogue entry, title "Twisted Groupoid Reconstruction of $C^*$-Algebra Pairs." The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is named.
• Supplied report citation, with mathematical typography normalized: $C^*$-Algebras (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/36, pp. 2059--2127. Original report: https://doi.org/10.4171/owr/2022/36. The extracted passage refers to "Theorem 1" without reproducing it. The registry separately records proposed year 2022.
• Supplied literature source: "Cartan subalgebras for non-principal twisted groupoid C-star algebras," https://arxiv.org/abs/2001.08270.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the combined entry open, reports that the report explicitly raises these three questions, and states that later work supplies sufficient cases without a complete answer to the combined entry. This is supplied dated curation, not independently verified here; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and states that it was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft.