Three-Cocycles in the Hyperfinite Type $\mathrm{II}_1$ Factor

Definitions and mathematical statement

Let $R$ be the usual hyperfinite type $\mathrm{II}_1$ factor with separable predual, let $\mathbb T=\{z\in{\mathbb{C}}:|z|=1\}$, and let $\operatorname{Out}(R)=\operatorname{Aut}(R)/\operatorname{Inn}(R)$. The source asks whether for every locally compact group $G$ and every class \[ [c]\in H^3_{\mathrm{Borel}}(G,\mathbb T) \] there exists an automorphic representation, or kernel, $\rho:G\to\operatorname{Out}(R)$ whose Connes--Jones invariant is $[c]$. The coefficient action on scalar $\mathbb T$ is trivial. The required continuity/measurability conventions and any countability assumption on $G$ are not specified.

In a lift convention making the obstruction explicit, choose measurable automorphisms $\alpha_g$ representing $\rho(g)$ and measurable unitaries $u(g,h)\in R$ with \[ \alpha_g\alpha_h=\operatorname{Ad}(u(g,h))\alpha_{gh}. \] Their associativity defect is the scalar $c(g,h,k)\in\mathbb T$ determined by \[ \alpha_g(u(h,k))u(g,hk) =c(g,h,k)u(g,h)u(gh,k). \] It is a $3$-cocycle; in the normalized convention its value is one if any argument is the identity, and it satisfies \[ c(h,k,l)c(g,hk,l)c(g,h,k) =c(gh,k,l)c(g,h,kl). \] Changing lifts changes the cocycle by a coboundary, leaving the obstruction class. Some pairing/order conventions replace this class by its inverse; the convention must match the cited invariant when comparing an individual class. Realizing a cohomology class does not require equality to a preselected cocycle at every triple.

The target is an outer automorphic representation with this possibly nonzero defect, not a genuine $G$-action lifted homomorphically to $\operatorname{Aut}(R)$, which has zero obstruction. No faithfulness condition is added. For general locally compact groups, pointwise Borel versus almost-everywhere measurable cohomology, measurable lifts and the topology on automorphisms must be fixed before the assertion is fully precise.

The supplied literature assessment checked 21 August 2026 reports realization of every third-cohomology class for countable discrete groups in the hyperfinite $\mathrm{II}_1$ factor, but says its located sources do not establish the full locally compact formulation. The discrete case is retained as partial progress, not generalized by forgetting the measurability issues.

Short English statement

Can every Borel third-cohomology class of a locally compact group occur as the scalar associativity obstruction of an outer representation on the hyperfinite $\mathrm{II}_1$ factor? The supplied assessment reports the countable discrete case, while the general group and measurability conventions remain unresolved here.

Sources

• Local UnsolvedMath registry, permanent ID 30000132, exact source code catalogue entry; Three-Cocycles in the Hyperfinite Type $\mathrm{II}_1$ Factor. Proposed year 2004, status partially_solved; corrected_verified statement identifies Borel cohomology and the Connes--Jones invariant.
• Supplied report citation: Nichtkommutative Geometrie (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/45, pp. 2349--2418, https://doi.org/10.4171/owr/2004/45. Title year differs from DOI/proposed year.
• Supplied literature assessment checked 2026-08-21: "Jones and subsequent obstruction results", https://people.maths.ox.ac.uk/cmi/library/proceedings/cmip011c.pdf; "Original OWR report", https://doi.org/10.4171/owr/2004/45; "Popa--Wassermann, compact Lie group actions", https://doi.org/10.1515/crll.2005.2005.578.147. This dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; linked PDFs were not accessed here.