TITLE: Lifting quasi-nilpotent elements in C$^*$-algebras
QUESTION [9 upvotes]: Let $A$ be a C$^*$-algebra with closed two-sided ideal $I$. Set $B=A/I$ and let $\pi:A\to B$ be the quotient map. Suppose that $b\in B$ is quasi-nilpotent. Does there exist quasi-nilpotent $a\in A$ such that $\pi(a)=b$?

REPLY [3 votes]: I am rather certain this is unknown.  The issue is that procedures we know to lift the relation $a^{n+1}=0$ do not mesh well with the $a^n=0$ case.  This means writing the given quasinilpotent as a limit of nilpotents does not seem to help.