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 $b^{n}=a$. 



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> To what extent such algebras are classified?
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A nilpotency question on $C^{*}$ algebras
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.


