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?

  • $\begingroup$ I suppose you aren't interested in commutative $A$'s? $\endgroup$ – Robert Israel Nov 12 '14 at 1:17
  • $\begingroup$ @RobertIsrael As you said, in this question we ignore the commutative $C^{*}$ algebras, since there is no nilpotent element other than $0$. $\endgroup$ – Ali Taghavi Nov 12 '14 at 1:23

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.

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