  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.
