5 problems

Jiang–Su stability and strict comparison conjecture. Having $\mathcal{Z}$-stability is equivalent to having strict comparison for positive elements.

Collapse conjecture. The algebra $\textsc{B}_\kappa$ will collapse $\kappa$ to $\omega_1$.

Let $A$ be a $\mathcal{Z}$-stable $C^*$-algebra, meaning that $A\cong A\otimes\mathcal{Z}$. Let $\operatorname{Cu}(A)$ be its Cuntz semigroup, and recall that a positively ordered…

Let $A$ be a simple unital nuclear separable $C^*$-algebra. The properties of finite nuclear dimension, $\mathcal{Z}$-stability, and strict comparison of positive elements are the…

Let $A$ be a separable, simple, unital, infinite-dimensional, nuclear $C^*$-algebra. The Toms' regularity conjecture. The following conditions are equivalent: 1. $A$ has finite nuc…