Cartan Subalgebras from Twisted Groupoid Subgroupoids

Definitions and mathematical statement

For a groupoid $G$, write $G^{(0)}$ for its unit space and $r,s:G\to G^{(0)}$ for the range and source maps. The groupoid is étale when $s$, equivalently $r$, is a local homeomorphism. A twist $\Sigma$ over $G$ is a central circle extension $G^{(0)}\times\mathbb T\hookrightarrow\Sigma\twoheadrightarrow G$, where $\mathbb T=\{z\in{\mathbb{C}}:|z|=1\}$. For a subgroupoid $S\subseteq G$, the restricted twist $\Sigma|_S$ is its inverse image in $\Sigma$. The subgroupoid is clopen if it is both open and closed in $G$.

Write $C_r^*(G;\Sigma)$ for the reduced twisted groupoid $C^*$-algebra, formed by completing the compactly supported sections of the associated complex line bundle in the reduced norm, whenever the intended topological and analytic hypotheses make this construction available. Put \[ A=C_r^*(G;\Sigma),\qquad B=C_r^*(S;\Sigma|_S). \] The intended inclusion $B\subseteq A$ is the canonical one induced by extension by zero of compactly supported sections on the open subgroupoid $S$.

A $C^*$-subalgebra $B\subseteq A$ is a Cartan subalgebra if all of the following hold:

1. $B$ is maximal abelian in $A$, meaning that $B$ is abelian and $B'\cap A=B$.
2. $B$ contains an approximate identity for $A$.
3. $B$ is regular: with \[ N_A(B)=\{a\in A:aBa^*\subseteq B\text{ and }a^*Ba\subseteq B\}, \] one has $\overline{\operatorname{span}}N_A(B)=A$.
4. There exists a faithful conditional expectation $E:A\to B$. Here a conditional expectation is a completely positive contractive projection onto $B$ satisfying $E(b_1ab_2)=b_1E(a)b_2$ for $a\in A$ and $b_1,b_2\in B$; faithfulness means $E(a^*a)=0$ implies $a=0$.

Requested characterization. For a twisted étale groupoid $(G,\Sigma)$, with no assumption of effectiveness, determine necessary and sufficient conditions on a clopen normal subgroupoid $S\subseteq G$, including its relation to the twist, for the canonical inclusion \[ C_r^*(S;\Sigma|_S)\subseteq C_r^*(G;\Sigma) \] to be a Cartan inclusion in the sense above. The original discussion identifies conditions forcing maximal abelianness as the central difficulty; it does not, in the supplied excerpt, establish the other Cartan requirements for every subgroupoid in the reconstructed statement.

Unresolved scope qualifications. The supplied record does not state the ambient local compactness, Hausdorffness, or countability hypotheses, or specify its convention for a normal subgroupoid. In particular, it does not explicitly say whether $S$ must contain all units or must lie in the isotropy $\operatorname{Iso}(G)=\{g\in G:r(g)=s(g)\}$. These requirements are not imposed here as additional source hypotheses. Thus the Cartan property is specified precisely, but the intended class of ambient groupoids and normal subgroupoids requires source clarification before a completely quantified characterization problem can be fixed. The original excerpt also asks about subgroupoids broadly, whereas the supplied reconstructed statement restricts to clopen normal subgroupoids and reduced twisted algebras.

Short English statement

Determine exactly when a clopen normal subgroupoid of a twisted étale groupoid gives a Cartan subalgebra of the ambient reduced twisted groupoid $C^*$-algebra. The ambient groupoid need not be effective. The intended topological hypotheses and meaning of normality remain unspecified, and the original excerpt has broader scope than the reconstructed statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005197$, title Cartan Subalgebras from Twisted Groupoid Subgroupoids, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record labels the statement corrected_verified and reports reconstruction and checking against the official report. This is supplied verification metadata, not an independent verification of this draft.
• Supplied report citation: $C^*$-Algebras (2023), Oberwolfach Reports, pp. 2059--2127, DOI 10.4171/owr/2022/36. Original report link: https://doi.org/10.4171/owr/2022/36. The record gives proposed year 2022; the relationship between that year and the citation's 2023 label is unresolved.
• Original discussion attribution: Anna Duwenig, Cartan subalgebras for non-effective twisted groupoid $C^*$-algebras, joint work with E. Gillaspy, R. Norton, S. Reznikoff, D. P. Williams, S. Wright, and J. Zimmerman. The record does not separately identify a proposer.
• Supplied literature source: Cartan subalgebras for non-principal twisted groupoid C-star algebras, https://arxiv.org/abs/2001.08270.
• Supplied literature source: Non-traditional C-star diagonals in twisted groupoid C-star algebras, https://arxiv.org/abs/2505.17913.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports sufficient dynamical and cocycle conditions, while reporting no established complete necessary-and-sufficient characterization of the broad question. This is supplied dated curation derived from the Oberwolfach record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.