Distances between approximate unitary equivalence classes of self-adjoints in C*-algebras

This thesis contains a study on a problem proposed by Toms, on whether the distance on the approximately unitary equivalence classes and the distance defined on morphisms from C[0, 1] to the Cuntz semigroup of a C*-algebra defined by Elliott and Ciuperca are equal on stable rank one C*-algebras. Previous research have shown that these distances define equivalent topologies on stable rank one C*-algebras. In this thesis we will explore on two different special cases of stable rank one C*-algebras, one simple and one non-simple. We will show that the two norms are equal on these two cases. Preliminaries and introduction are in Chapter 1 and 2 while we deal with the special case for simple unital exact Z -stable C*-algebra in Chapter 4. Finally the proof that the two norms are equal on inductive limits of 1-dimensional NCCW complexes will be shown in Chapter 5. Further research direction will be given in Chapter 6. Furthermore in this thesis, we consider whether these two distances are characterized by the spectral information of the positive elements that represent the class. In particular, we show that the two distances are equal to the distance between spectra of operators for the first case and for simple unital inductive limits of 1-dimensional NCCW complexes.