Ultraproduct II_1 Factors — Problem 46

Definitions and mathematical statement

A type $\mathrm{II}_1$ factor is a von Neumann algebra $M$ with center $Z(M)={\mathbb{C}} 1$, no nonzero minimal projections, and a faithful normal tracial state $\tau$ satisfying $\tau(1)=1$. Its underlying unital $C^*$-algebra has the same multiplication, involution, unit, and operator norm.

The source asks: Is every separable type $\mathrm{II}_1$ factor selfless as a $C^*$-algebra?

Under the usual von Neumann algebra convention that separability means norm separability of the predual $M_*$, the requested assertion would be \[ \forall M\;\left[\left(M\text{ is a type }\mathrm{II}_1\text{ factor and }M_*\text{ is norm separable}\right)\Longrightarrow \mathsf{Selfless}(M)\right], \] where $\mathsf{Selfless}(M)$ denotes the source's intended property of the underlying $C^*$-algebra. This notation is a placeholder, not a definition: the supplied record does not define selfless or give its defining quantifiers and hypotheses. The record also does not explicitly specify the separability convention. Consequently, the displayed formulation is conditional on that convention, and a complete mathematical statement requires the intended definition of selflessness.

Short English statement

Elayavalli's Problem 46 asks whether every separable type $\mathrm{II}_1$ factor, regarded as a $C^*$-algebra, is selfless. The supplied record omits the definition of selflessness and does not explicitly specify the separability convention.

Sources

• Local UnsolvedMath registry, permanent ID $801144$, source code catalogue entry, title Ultraproduct $\mathrm{II}_1$ Factors --- Problem 46; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: Elayavalli (2025), 50 Open Problems: Ultraproduct $\mathrm{II}_1$ Factors, Problem 46. Supplied source URL: https://arxiv.org/abs/2511.20377. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not independently verified access or verification.
• The registry labels the problem open. This is a supplied status label, not an independently verified mathematical or literature assessment.