Generalized Group Actions for Crossed-Product Operator Algebras

Definitions and mathematical statement

Let $A$ be a $C^*$-algebra, and let $\operatorname{Aut}(A)$ denote its group of $*$-automorphisms. At the algebraic level, an action of a group $G$ on $A$ is a homomorphism $\alpha\colon G\to\operatorname{Aut}(A)$. When a topology on $G$ is involved, additional continuity hypotheses must be specified; the supplied question does not specify them. A crossed product associated with an ordinary action is customarily denoted by $A\rtimes_\alpha G$, with the particular crossed-product construction requiring further conventions.

Source question. How can group actions be generalized so that a crossed-product $C^*$-algebra can still be defined?

The requested objective is to identify a notion of generalized action and an associated crossed-product construction extending an ordinary group-action construction. The record does not specify the admissible groups, generalized acting objects, coefficient algebras, continuity conditions, full or reduced crossed-product convention, or a universal property characterizing the extension. Consequently, it does not determine a single quantified mathematical assertion.

Unresolved source qualification. The supplied statement-verification note says that the contribution immediately answers its motivating question by identifying Fell bundles as the appropriate generalized actions. It therefore promotes no separate clean statement and labels the statement unrecoverable. The supplied broader literature assessment also describes the question as having no unique complete answer. Thus this draft preserves a research question and its qualifications; it does not assert an unresolved existence theorem or formulate a new conjecture. The reference to Cuntz--Pimsner algebras occurs in the contribution title and supplies no separate mathematical problem.

Short English statement

The source asks which generalizations of group actions admit crossed-product $C^*$-algebras. Its supplied verification note reports that the contribution answers this motivating question using Fell bundles. Because the intended scope and defining properties are unspecified, the record does not yield a precise standalone conjecture. The supplied open label is retained as registry metadata, not as an independently verified mathematical status.

Sources

• Local UnsolvedMath registry: permanent ID $30005191$, source problem code catalogue entry, title "Generalized Group Actions for Crossed-Product Operator Algebras," in the Oberwolfach Reports Open Problems collection. The supplied record has status open, statement status unrecoverable, and no clean statement.
• Supplied report citation: Non-Commutative Geometry and Cyclic Homology (2023), Oberwolfach Reports, pp. 1987--2058, DOI 10.4171/owr/2022/35. Original Oberwolfach report: https://doi.org/10.4171/owr/2022/35. The registry separately supplies proposed year $2022$; the relationship between that year and the citation's $2023$ designation is not resolved here.
• Supplied contribution attribution: Ralf Meyer, "A bicategorical interpretation for crossed products and Cuntz--Pimsner algebras," joint work with Suliman Albandik, Alcides Buss, Camila Fabre Sehnem, and Chenchang Zhu. No separate proposer is specified.
• Supplied literature source: Colimits in the correspondence bicategory, https://arxiv.org/abs/1502.07771.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified. It labels the problem open, reports generalizations involving groupoids, Fell bundles, crossed modules, correspondences, and related structures, and says that the broad extracted question has no unique complete answer. The background explicitly requests independent verification of time-sensitive status and cited claims. The supplied statement-verification note separately reports that the contribution answers its motivating question using Fell bundles; neither source has been independently consulted for this draft.