
8. Generators

1.     A C*-algebra is called singly generated if it contains an element that is not contained in any proper sub-C*-algebra. A C*-algebra is called $\mathcal{Z}$-stable if it absorbs the Jiang-Su 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?
It is known that separable, unital C*-algebras are singly generated if they absorb the Jiang-Su algebra.

Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.