Nonembeddable Tracial Algebras with Wasserstein Lifting

Definitions and mathematical statement

A tracial von Neumann algebra is denoted by $(M,\tau)$, where $M$ is a von Neumann algebra and $\tau$ is its specified trace. The supplied record does not explicitly state the normalization, faithfulness, or normality requirements on $\tau$, or any separability or generation hypothesis on $M$.

Connes embeddability ordinarily concerns a trace-preserving unital $*$-embedding into a tracial ultrapower of the hyperfinite type $\mathrm{II}_1$ factor. The record does not specify the ultrapower conventions or the size hypotheses under which this terminology is intended. These conventions require confirmation from the report.

For a specified finite self-adjoint tuple $x=(x_1,\ldots,x_d)$ in $M$, its noncommutative law is the functional \[ \mu_x(p)=\tau\left(p(x_1,\ldots,x_d)\right),\qquad p\in{\mathbb{C}}\langle X_1,\ldots,X_d\rangle. \] This definition explains the terminology; it does not impose a finite generation hypothesis on the requested algebra. The source record supplies neither a tuple nor the space of laws on which the two topologies are to be compared.

Source-qualified existence question. Does there exist a non-Connes-embeddable tracial von Neumann algebra $(M,\tau)$ satisfying both of the following requirements?

1. Its noncommutative law has coincident weak-star and Wasserstein topologies, in the sense intended by the source report.
2. Every ultraproduct embedding of the algebra lifts to factorizable completely positive maps, in the sense intended by the source report.

The universal quantifier in the second requirement is retained. The supplied original passage states only that it is an open question whether non-Connes-embeddable tracial $W^*$-algebras satisfying "(1) and (2)" exist; it does not reproduce those numbered conditions. The two requirements above come from the supplied reconstructed statement, and their identification with the report's numbered conditions cannot be checked from this record alone.

Unresolved formulation. Equality of topologies requires a specified underlying space; the record does not determine whether equality on a space of laws or agreement of convergence at a particular law is intended. It also omits the Wasserstein distance, moment or operator-norm bounds, and the tuple or generating family involved. For the lifting requirement, the record omits the class of target algebras, the ultrafilter and index set, the trace requirements on embeddings, the meaning of a lift, and the convention for factorizability, including any unitality and trace-preservation requirements. Consequently, the displayed question preserves the supplied target but is not a fully determined mathematical proposition until these qualifications are recovered.

Short English statement

The supplied question asks for a tracial von Neumann algebra that is not Connes embeddable and satisfies both a weak-star/Wasserstein topology condition and a universal lifting condition for ultraproduct embeddings. The exact conditions are missing from the quoted report passage, so their domains and quantifiers cannot be fully specified from this record. The supplied open status and verification labels require independent human review.

Sources

• Local UnsolvedMath problem registry, permanent ID $30005198$, exact source problem code catalogue entry, title Nonembeddable Tracial Algebras with Wasserstein Lifting; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2022.
• Supplied report citation: $C^*$-Algebras (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/36, pp. 2059--2127. Original Oberwolfach report: https://doi.org/10.4171/owr/2022/36. The title's 2023 date is retained alongside the record's 2022 proposed year and DOI year; their relationship is not resolved here. The supplied title has malformed dollar-sign markup, normalized here for LaTeX presentation.
• Supplied literature source: Duality for optimal couplings in free probability, https://doi.org/10.1007/s00220-022-04480-0. No authors or pinpoint location are supplied in the record.
• Literature assessment checked at 2026-08-22: the supplied curation says that non-Connes-embeddable tracial von Neumann algebras are known to exist, while existence with the additional conditions (1) and (2) remains open. This is supplied dated curation, not an independently verified assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft. The registry's problem status is open.