Closed unitary and similarity orbits of normal operators in purely infinite C⁎-algebras

We will investigate the norm closure of the unitary and similarity orbits of normal operators in unital, simple, purely infinite C⁎-algebras. A simple proof to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial K1-group will be given using previously known techniques. Using these techniques some upper and lower bounds for the distance between unitary orbits of normal operators will be obtained. In addition, a complete characterization of when one normal operator is in the closed similarity orbit of another normal operator will be given for unital, simple, purely infinite C⁎-algebras and type III factors with separable predual.