Zack Cramer, Pure Mathematics, University of Waterloo
"Approximation of Normal Operators by Nilpotents in Purely Infinite $C^*$-algebras"
A famous problem posed by Halmos in 1970 asked whether every quasinilpotent operator (that is, an operator whose spectrum is $\{0\}$) on a complex, separable, infinite-dimensional Hilbert space is a norm limit of nilpotent operators. An affirmative answer was obtained in 1974, when Apostol, Foia\c{s} and Voiculescu provided a complete characterization of the closure of nilpotent operators in this setting. Recently, Skoufranis derived a similar characterization of the normal limits of nilpotents in the case of a unital, simple, purely infinite $C^*$-algebra.
We will look briefly at the techniques used in proof of the classical case and provide an overview of the extensions made by Skoufranis. In particular, we shall see why the algebras considered by Skoufranis are good candidates for generalizing the solution to Halmos' problem, and provide insight into possible directions that could extend these results further.