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Let $A$ be a C$^*$-algebra. I have seen theorems either stating or implying that if $A$ is the algebra of bounded linear operators on a separable Hilbert space (Herrero et al.), or the Calkin algebra, or an AF-C$^*$-algebra (Skoufranis), then every quasi-nilpotent element in $A$ is a norm-limit of nilpotent elements. Is the general case known: is every quasi-nilpotent element in a C$^*$-algebra a norm-limit of nilpotent elements?

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