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. An operator theoretic proof will be given to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial K_1-group. Some upper and lower bounds for the distance between unitary orbits will be obtained based on these methods. 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.