Dini Spaces as Primitive Spectra of Amenable $C^*$-Algebras

Definitions and mathematical statement

Let $\mathcal D_{\mathrm{source}}$ denote the class of topological spaces called Dini spaces in the supplied source. The record does not define this term or specify its topological axioms. Consequently, the statement below preserves the source terminology; a self-contained characterization of its quantified domain remains unresolved.

A complex $C^*$-algebra $A$ is separable if it has a countable norm-dense subset. Interpret amenability in its standard $C^*$-algebra sense, equivalently nuclearity: for every finite subset $F\subseteq A$ and every $\varepsilon>0$, there are an integer $n\geq 1$ and completely positive contractive linear maps \[ A\xrightarrow{\phi}M_n({\mathbb{C}})\xrightarrow{\psi}A \] such that $\|\psi(\phi(a))-a\|<\varepsilon$ for all $a\in F$. A linear map is completely positive if all its matrix amplifications preserve positivity, and contractive if its norm is at most one.

The primitive spectrum of $A$ is \[ \operatorname{Prim}(A)=\{\ker\pi:\pi\text{ is a nonzero irreducible }*\text{-representation of }A\}. \] Here irreducibility means that the representation has no proper nonzero closed invariant subspace. Equip $\operatorname{Prim}(A)$ with the hull--kernel topology, whose closed sets are \[ \operatorname{hull}(I)=\{P\in\operatorname{Prim}(A):I\subseteq P\}, \] where $I$ ranges over norm-closed two-sided ideals of $A$.

Question. Is it true that \[ \forall X\in\mathcal D_{\mathrm{source}}\quad \exists\, A\text{ a separable amenable complex }C^*\text{-algebra}\quad X\cong\operatorname{Prim}(A), \] where $\cong$ denotes a homeomorphism? The record states no unitality requirement on $A$. Any boundary conditions or separation assumptions implicit in the term Dini space must be recovered from the source before this becomes a fully specified standalone statement.

Short English statement

The question asks whether every Dini space can occur, up to homeomorphism, as the primitive spectrum of a separable amenable $C^*$-algebra. The supplied record labels this question open, but does not define Dini space. Its exact topological domain therefore requires source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002395$, Oberwolfach Reports Open Problems collection, exact source problem code catalogue entry. No individual proposer is supplied; the recorded proposed year is 2013.
• Supplied source citation: C*-Algebren (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/43, pp. 2423--2500. https://doi.org/10.4171/owr/2013/43. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions. The citation's 2014 label and the 2013 year appearing in the DOI and proposed-year field are retained without reconciliation.
• Supplied literature reference: Harnisch--Kirchberg primary paper, https://arxiv.org/abs/2401.05917.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and states that the 2024 inverse-problem work characterizes realizable primitive spectra topologically, without showing that every Dini space satisfies that characterization. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports a comparison with the official Oberwolfach report and normalization of notation, context, and question form without changing the source-level claim. That verification is reported provenance and has not been independently checked for this draft.