| Summary: | <p>In this thesis, we will use classification results for C<sup>∗</sup>-algebras and <sup>∗</sup>-homomorphisms between them to characterise nuclear dimension equal to zero for a large class of <sup>∗</sup>-homomorphisms. In particular, for certain <sup>∗</sup>-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 <sup>∗</sup>-homomorphisms between commutative C<sup>∗</sup>-algebras, we develop a notion of real rank zero for inclusions of C<sup>∗</sup>-algebras. Among others, we provide interesting examples from dynamics that have this property and show that full <em>O</em><sub>∞</sub>-stable inclusions have real rank zero.</p>
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<p>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<sup>∗</sup>-algebras with real rank zero.</p>
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<p>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<sup>∗</sup>-algebras that have a faithful amenable trace.</p>
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