The corona factorization property and the Cuntz semigroup in the classification of C*-algebras

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In 1989, George Elliott proposed that separable amenable C*-algebras could be completely classified by their K-theory and traces. While the verification was very successful in the following years, counterexamples were found eventually. Hence, for a complete classification, an extension of the invariant is necessary. A candidate for an extended invariant is the Cuntz semigroup, but it remains challenging to work with it in the classification program.I aim to study this new invariant, and to further study the algebras that prove the original Elliott invariant to be insufficient.The results of my thesis arose from trying to understand the correlation of two specific regularity properties for (non-classifiable) C*-algebras. These two properties hold for all C*-algebras for which classification by the original Elliott invariant is possible. Also counterexamples to Elliott's conjecture can have these two properties, but certain exotic behavior is ruled out by them. To better understand the two regularity properties and their correlations requires developments of existing techniques, which would be beneficial in the general program of C*-algebra classification.The chosen supervisor matches the interests of the researcher and has several publications in the field of study. His publications include the justification to consider the Cuntz semigroup as an additional invariant, which reconciles the principle of Elliott's classification. Also his publications provide promising tools for the research project.