Classification Complexity of Unitary Equivalence

Definitions and mathematical statement

A $C^*$-algebra $A$ is separable if its norm topology has a countable dense subset. For equivalence relations $E$ on a standard Borel space $X$ and $F$ on a standard Borel space $Y$, write $E\leq_B F$ if there is a Borel map $f:X\to Y$ such that \[ x\mathrel{E}x'\quad\Longleftrightarrow\quad f(x)\mathrel{F}f(x') \qquad(x,x'\in X). \] The relation $E$ is smooth if $E\leq_B {=}_{{\mathbb{R}}}$. It is classifiable by countable structures if there exists a countable first-order language $L$ such that $E\leq_B\cong_L$, where $\cong_L$ denotes isomorphism on the standard Borel space $\operatorname{Mod}_{{\mathbb{N}}}(L)$ of $L$-structures with underlying set ${\mathbb{N}}$.

The supplied question asks whether, for every separable $C^*$-algebra $A$, \[ E_A^{\mathrm{u.e.}}\not\leq_B {=}_{{\mathbb{R}}} \quad\Longrightarrow\quad \text{for every countable language }L,\quad E_A^{\mathrm{u.e.}}\not\leq_B\cong_L. \] Here $E_A^{\mathrm{u.e.}}$ is called a unitary-equivalence relation in the supplied statement. However, the record does not specify the objects being compared, their standard Borel parameter space, or the precise meaning of unitary equivalence. The displayed implication is therefore a formulation schema, conditional on supplying those definitions, and is not a fully determined mathematical assertion. In particular, the linked reference concerning automorphisms does not by itself identify the intended domain. No unitality or continuous-trace hypothesis is stated in the question.

Short English statement

The question asks whether every nonsmooth unitary-equivalence relation associated with a separable $C^*$-algebra fails to admit classification by countable structures through Borel maps. The record leaves the relation and its parameter space unspecified, so its intended assertion requires clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002221$, entitled Classification Complexity of Unitary Equivalence; Oberwolfach Reports Open Problems collection, exact source code catalogue entry. The record gives proposed year 2012 and names no proposer.
• Supplied report citation: C*-Algebras, Dynamics, and Classification (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/52, pp. 3129--3209. Source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2012/52. This report is supplied as the authoritative provenance; its contents have not been independently checked here.
• Supplied literature-source link labelled "unitary equivalence of automorphisms": https://arxiv.org/abs/1304.3502. The record supplies no authors or more detailed bibliographic attribution for this link.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open, says that the isolated symbol $E_A$ is undefined in the extracted record, and reports related nonclassifiability results for non-continuous-trace algebras and smoothness in a unital continuous-trace case. It also states that the intended general assertion cannot be verified and requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review describing the cleanup as self-contained and meaning-preserving. These supplied review labels do not resolve the missing definition of the relation or establish mathematical correctness.