Closed Unitary and Similarity Orbits of Normal Operators in Purely Infinite C*-Algebras

TL;DR

This paper studies the approximation and classification of normal operators in purely infinite C*-algebras, providing new criteria for their unitary and similarity orbit closures and their approximate equivalences.

Contribution

It offers a novel operator theoretic approach to classify normal operators' approximate unitary equivalence and characterizes when one is in the similarity orbit of another in these algebras.

Findings

Classification of approximate unitary equivalence for normal operators

Bounds for distances between unitary orbits

Complete characterization of similarity orbit containment

Abstract

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.