9 problems

Blackadar–Handelman conjecture. The set

Let $A$ be a unital, separable, nuclear ${\rm C}^*$-algebra of real rank zero and stable rank one, and let $\underline{\mathrm{Cu}}_u(A)$ denote its total Cuntz semigroup invariant…

Let $A$ be a simple, unital $C^*$-algebra. Say that all ranks are realized if, for every lower-semicontinuous affine function $f\colon QT(A)\to(0,\infty]$, there exists…

Cuntz semigroup classification conjecture. The functor $\operatorname{Cu}$ is a classifying functor for unitary elements of any unital $\operatorname{AF}$-algebra; equivalently, it…

Let $A$ be a stably finite unital C-algebra and let $X$ be a compact metric space. The radius of comparison, denoted by ${\operatorname{rc}}$, measures comparison in the Cuntz semi…

Let $X$ be a compact metric space and let $h$ be a minimal homeomorphism of $X$. Denote by $C^*({\mathbb Z},X,h)$ the crossed product associated to the induced action of…

Round ideal completion conjecture. For all $a,b\in P$,

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…

Classification conjecture. Simple separable amenable $C^*$-algebras should be classified up to $*$-isomorphism by the Elliott invariant together with the Cuntz semigroup.