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 $\pi(a)=b$?



Lifting quasi-nilpotent elements in C$^*$-algebras
If I is the compact elements of A and B is the corresponding Calkin Algebra, the answer is yes. Have you looked at BARNES, B. A., MURPHY, G. J., SMYTH, M. R. F. and WEST, T. T., "Riesz and Fredholm theory in Banach algebras" (Research Notes in Mathematics 67, Pitman, 1982?


