TITLE: A nilpotency question on $C^{*}$ algebras
QUESTION [5 upvotes]: What is  an example of a $C^{*}$  algebra $A$ with the property that: for  every nilpotent(Quasi nilpotent) $a$ and for  every $n\in \mathbb{N}$, there is a  $b$  with $b^{n}=a$. 

To what extent such  algebras are  classified?

REPLY [3 votes]: I'm not sure if this is useful: If your $C^*$-algebra also has a non-zero nilpotent element, then it will have nilpotent elements of all orders, it is not (algebraically) of bounded index, so does not satisfy a polynomial identity.  $C^*$-algebras which do satisfy a polynomial identity are the subject of the following paper: B. E. Johnson. "Near inclusions for subhomogeneous $C^*$-algebras", Proc. London Math. Soc., 68:399–422, 1994.