Dynamical Reconstruction from Bare Crossed Product Banach Algebras

Definitions and mathematical statement

Retain the source notation $K$, $X_K$, $I_K$, $A_K$, and $\sigma_K$. The supplied record describes an action of $I_K$ on $X_K$, a diagonal algebra denoted by $C(X_K)$, and the algebraic crossed product $C(X_K)\rtimes_{\mathrm{alg}} I_K$. Write \[ B_K:=\overline{\rho_K\left(C(X_K)\rtimes_{\mathrm{alg}} I_K\right)}^{\|\cdot\|_{\mathrm{op}}}, \] where $\rho_K$ denotes the representation called the natural bounded-operator representation in the source. This notation records the stated closure construction; the supplied record does not specify $\rho_K$ or the crossed-product conventions. The algebra $B_K$ is considered as a non-involutive Banach algebra. The source calls $C(X_K)$ a MASA, meaning a maximal abelian self-adjoint subalgebra, but does not specify its ambient involutive algebra. The symbol $\sigma_K$ denotes the time evolution mentioned in the supplied background.

1. Can the dynamics consisting of $I_K$ acting on $X_K$ be reconstructed from $B_K$?
2. Can this reconstruction be achieved without requiring preservation of the distinguished MASA $C(X_K)$?
3. Can $K$ be reconstructed from $A_K$ without using $\sigma_K$?

These are the reconstruction questions supplied by the record. A fully quantified reconstruction assertion cannot yet be stated: the record does not define the admissible $K$, the constructions of $X_K$, $I_K$, and $A_K$, the relation between $A_K$ and $B_K$, the allowed algebra isomorphisms, or the equivalence relation under which dynamics and fields are to be recovered. In particular, the background describes field reconstruction but does not specify the class of fields. These choices remain unresolved and must not be inferred from the notation alone.

The supplied dated curation labels the problem partially solved and reports reconstruction results when the relevant diagonal algebra or time evolution is respected. It describes reconstruction from the bare Banach algebra without preserving the MASA, and reconstruction from $A_K$ without $\sigma_K$, as remaining open in the extracted formulation. These are supplied literature assessments, not independently verified mathematical conclusions.

Short English statement

The questions ask whether the crossed-product Banach algebra determines the action $I_K\curvearrowright X_K$, including when its diagonal $C(X_K)$ is not required to be preserved, and whether $A_K$ determines $K$ without its time evolution $\sigma_K$. The underlying constructions and the intended meanings of reconstruction are missing, so this remains a qualified statement-only draft.

Sources

• Local UnsolvedMath registry, supplied complete source record for permanent ID $30003047$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Title: Dynamical Reconstruction from Bare Crossed Product Banach Algebras. The record gives the proposed year as 2016 and supplies no individual proposer attribution.
• Supplied report citation: Mini-Workshop: Operator Spaces and Noncommutative Geometry in Interaction (2016), Oberwolfach Reports, pp. 267--295, DOI 10.4171/owr/2016/6. Source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2016/6. No problem-specific page or passage is supplied.
• Supplied literature-source link labeled "Cornelissen--Marcolli QSM and anabelian geometry": https://arxiv.org/abs/1009.0736. The record supplies no theorem number or precise correspondence between this reference and the individual reconstruction questions.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved, reports reconstruction when the relevant diagonal algebra or time evolution is respected, and describes the bare-algebra questions as open in the extracted formulation. This is supplied dated curation derived from the OWR material, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, with the qualification that only safe notation, typography, and TeX-delimiter cleanup was applied and that hypotheses, quantifiers, and subquestions were preserved. This editorial label does not establish mathematical correctness or resolve the missing definitions.