Scope of the Generalized Tracial-Rank-One Class

Definitions and mathematical statement

Work with complex $C^*$-algebras. A $C^*$-algebra is separable if it has a countable norm-dense subset, unital if it has a multiplicative identity, and simple if it is nonzero and has no nonzero proper closed two-sided ideals. Interpret amenability in its usual $C^*$-algebraic sense, equivalently nuclearity. Let $\mathcal Z$ denote the Jiang--Su algebra; $\mathcal Z$-stability means $A\cong A\otimes_{\min}\mathcal Z$ as $C^*$-algebras. The UCT condition means that $A$ satisfies the universal coefficient theorem for $KK$-theory.

A UHF algebra is a unital inductive limit of full matrix algebras with unital connecting maps. It is of infinite type if its supernatural number $\prod_p p^{n_p}$ has $n_p\in\{0,\infty\}$ for every prime $p$, with at least one exponent equal to $\infty$. Write $\operatorname{gTR}(B)$ for the generalized tracial rank of $B$. The supplied record names this technical invariant but does not provide its defining approximation class or detailed convention; these must be checked against the cited report before a fully self-contained definition can be fixed.

Using that source-dependent generalized-tracial-rank predicate, define \[ \mathcal N_1=\left\{A\;\middle|\; \begin{gathered} A\text{ is a separable, unital, simple, amenable }C^*\text{-algebra},\\ A\cong A\otimes_{\min}\mathcal Z,\qquad A\text{ satisfies the UCT},\\ \exists\,U\text{ an infinite-type UHF algebra such that} \quad\operatorname{gTR}(A\otimes_{\min}U)\leq 1 \end{gathered} \right\}. \] The existentially quantified algebra $U$ may depend on $A$. The bound is at most one, so the definition does not require generalized tracial rank to equal one.

Problem. Determine the scope of $\mathcal N_1$ and characterize which $C^*$-algebras belong to it. The supplied formulation asks both how large this class is and which algebras it contains, but specifies neither a notion of size nor a particular proposed membership criterion. Thus it is a characterization problem rather than a single stated conjectural equivalence. The UCT hypothesis is included in the supplied corrected statement, although it is absent from the supplied original wording.

Short English statement

Describe the class $\mathcal N_1$ of separable, unital, simple, amenable, $\mathcal Z$-stable $C^*$-algebras satisfying the UCT whose generalized tracial rank becomes at most one after tensoring with some infinite-type UHF algebra. Determine its extent and membership criteria. The intended meaning of size and the detailed generalized-tracial-rank convention require clarification from the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003201$, title: Scope of the Generalized Tracial-Rank-One Class; source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied report citation, with mathematical title markup normalized: $C^*$-Algebras (2017), Oberwolfach Reports, pp. 2269--2345, DOI 10.4171/owr/2016/40. https://doi.org/10.4171/owr/2016/40. The displayed year 2017 and the 2016 year in the DOI and proposed-year metadata are retained without reconciliation.
• Supplied literature source: OWR Operator Algebras report. https://ems.press/content/serial-article-files/46645?nt=1.
• Supplied literature source: Gardella and Perera Cuntz semigroups survey. https://doi.org/10.4171/EMSS/84.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports strong classification and realization theorems for $\mathcal N_1$, while stating that a complete intrinsic description of its members is not known. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked to restore the UCT hypothesis, tensor factor, generalized-tracial-rank condition, and both scope questions. This verification claim belongs to the supplied record; the cited sources have not been independently consulted for this draft.