Central Sequence Invariants for $C^*$-Algebra Classification

Definitions and mathematical statement

The supplied record concerns a $C^*$-algebra $A$ and two invariants denoted by $F(A)$ and $F_{\mathrm{cont}}(A)$. It does not define either invariant or specify the class of $C^*$-algebras under consideration. These symbols are therefore retained as source notation, without assigning them constructions unsupported by the record.

For a $C^*$-algebra $D$, write $K_0(D)$ and $K_1(D)$ for its topological $K$-groups. Whenever the relevant Kasparov groups and products are defined, $A$ and $B$ are $KK$-equivalent if there are elements $x\in KK(A,B)$ and $y\in KK(B,A)$ satisfying \[ x\otimes_B y=1_A\in KK(A,A),\qquad y\otimes_A x=1_B\in KK(B,B). \]

Source conjecture, with unresolved qualifications. The invariant $F(A)$ contains all technical information needed to decide whether $A$ is classifiable, and some information about its class. The $K$-groups of $F_{\mathrm{cont}}(A)$ determine the $KK$-equivalence class of $A$.

The first assertion cannot be turned into a precise predicate from this record: the meaning of "classifiable," the classification framework, the information retained by $F(A)$, and the intended information about the class of $A$ are unspecified. In particular, "some information about its class" does not assert that $F(A)$ completely determines the isomorphism class of $A$.

For the second assertion, let $\mathcal D$ stand for the intended, but unspecified, domain of algebras. If the intended comparison is by isomorphisms of the two ordinary $K$-groups preserving their degrees, its determination claim would have the schematic form \[ \forall A,B\in\mathcal D,\qquad \left[ K_i\!\left(F_{\mathrm{cont}}(A)\right) \cong K_i\!\left(F_{\mathrm{cont}}(B)\right) \text{ for }i=0,1 \right] \Longrightarrow A\simeq_{KK}B. \] This display is conditional on that interpretation. The record does not establish the domain $\mathcal D$, the construction of $F_{\mathrm{cont}}$, or whether additional structure on the $K$-groups must be preserved. A fully specified quantified conjecture therefore requires further source clarification.

Short English statement

The supplied conjecture asks whether $F(A)$ supplies the information needed to recognize classifiable $C^*$-algebras, together with some information about their classes, and whether the $K$-groups of $F_{\mathrm{cont}}(A)$ determine $A$ up to $KK$-equivalence. The definitions, domain, and classification framework are missing, so the formulation remains qualified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002227$, exact source problem code catalogue entry, titled "Central Sequence Invariants for $C^*$-Algebra Classification." The collection is Oberwolfach Reports Open Problems. The proposed year is recorded as 2012; no individual proposer is supplied.
• Supplied report citation: C*-Algebras, Dynamics, and Classification (2013), Oberwolfach Reports, pp. 3129--3209, DOI 10.4171/owr/2012/52. https://doi.org/10.4171/owr/2012/52. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Oberwolfach report, Conjecture 2": https://publications.mfo.de/bitstream/handle/mfo/3323/OWR_2012_52.pdf?isAllowed=y&sequence=1.
• Supplied literature source, labeled "Kirchberg--Rordam, central sequence algebras": https://arxiv.org/abs/1209.5311.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the proposed completeness of $F(A)$ and $F_{\mathrm{cont}}(A)$ as conjectural; it states that known structural facts do not prove that these invariants decide classification or $KK$-equivalence. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.