Structural Problems for $C^*$- and von Neumann Algebras

Definitions and mathematical statement

Shared notation.

$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.

Statement recorded in the supplied source.

Do the normalized random matrices $X_k$ converge strongly to a free circular distribution? Are $q$-Gaussian $C^*$-algebras selfless for $-1<q<1$ and $q\neq0$? Are the indicated tracial ultrapowers mutually isomorphic? Does there exist a group that has a mixed identity and whose reduced $C^*$-algebra is selfless? Does $C^*(F_2\times F_2)$ have the local lifting property? Does a nonamenable group without free subgroups exist whose reduced $C^*$-algebra has rapid decay? For compact convex $K$, does equality of its minimal and maximal operator-system structures imply that $K$ is a Choquet simplex? Are the free group factors $L(F_2)$ and $L(F_3)$ nonisomorphic? Are the free-product factors $Z(r,\tau)^{*n}$ and $Z(r,\tau)^{*m}$ isomorphic, and is there a dichotomy? Is every unital simple separable $C^*$-algebra singly generated? Do classifiable $C^*$-algebras admit filtrations with rapid decay? Is $F_2$ superrigid among hyperbolic groups at the level of reduced $C^*$-algebras? When is a von Neumann algebra with a faithful state selfless?

Formulation references: [S1], [S2].

Short English statement

Do the normalized random matrices $X_k$ converge strongly to a free circular distribution? Are $q$-Gaussian $C^*$-algebras selfless for $-1<q<1$ and $q\neq0$? Are the indicated tracial ultrapowers mutually isomorphic? Does there exist a group that has a mixed identity and whose reduced $C^*$-algebra is selfless? Does $C^*(F_2\times F_2)$ have the local lifting property? Does a nonamenable group without free subgroups exist whose reduced $C^*$-algebra has rapid decay? For compact convex $K$, does equality of its minimal and maximal operator-system structures imply that $K$ is a Choquet simplex? Are the free group factors $L(F_2)$ and $L(F_3)$ nonisomorphic? Are the free-product factors $Z(r,\tau)^{*n}$ and $Z(r,\tau)^{*m}$ isomorphic, and is there a dichotomy? Is every unital simple separable $C^*$-algebra singly generated? Do classifiable $C^*$-algebras admit filtrations with rapid decay? Is $F_2$ superrigid among hyperbolic groups at the level of reduced $C^*$-algebras? When is a von Neumann algebra with a faithful state selfless?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006427 (OWR-14299522-011). Catalogue attribution under CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Original problem formulation and mathematical context. https://doi.org/10.4171/owr/2025/35.