8. Generators

A C*algebra is called singly generated if it contains an element that is not contained in any proper subC*algebra. A C*algebra is called $\mathcal{Z}$stable if it absorbs the JiangSu algebra tensorially.
Problem 8.1.
Which separable, unital, simple C*algebras are singly generated? Is there a simple, nuclear C*algebra that is singly generated, yet is not $\mathcal{Z}$stable? Is single generation connected to the regularity properties coming up in the classification of nuclear, simple C*algebras?
Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.