Applications of classification of C*-algebras

Thesis

In this thesis, we will use classification results for C^∗-algebras and ^∗-homomorphisms between them to characterise nuclear dimension equal to zero for a large class of ^∗-homomorphisms. In particular, for certain ^∗-homomorphisms where the codomain is a sequence algebra, having nuclear dimension equal to zero is equivalent to factoring through a simple AF-algebra. As a byproduct of characterising nuclear dimension equal to zero for ^∗-homomorphisms between commutative C^∗-algebras, we develop a notion of real rank zero for inclusions of C^∗-algebras. Among others, we provide interesting examples from dynamics that have this property and show that full O_∞-stable inclusions have real rank zero.

We further use classification results for automorphisms of A𝕋-algebras of real rank zero to build flows with specified KMS behaviour on all unital UCT Kirchberg algebras. In the tracial case, we obtain similar results for all finite classifiable C^∗-algebras with real rank zero.

In the last chapter, we use classification techniques involving the trace-kernel extension to show that the property that all amenable traces are quasidiagonal is invariant under homotopy for separable exact C^∗-algebras that have a faithful amenable trace.

Authors: Neagu, R-M

• DOI: 10.5287/ora-2zazvdnew
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: KMS states; C*-algebras; classification of C*-algebras; Homomorphisms (mathematics)