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

1989年，George Elliott提出可分可服从C*-代数可以通过其k理论和迹完全分类。虽然在接下来的几年里验证非常成功，但最终还是发现了反例。因此，对于一个完整的分类，不变量的扩展是必要的。扩展不变量的候选是Cuntz半群，但在分类程序中使用它仍然具有挑战性。我的目的是研究这个新的不变量，并进一步研究证明原来的Elliott不变量是不充分的代数。我的论文结果源于试图理解（不可分类的）C*-代数的两个特定正则性的相关性。这两个性质适用于所有可以用原始的艾略特不变量分类的C*-代数。艾略特猜想的反例也可以有这两种性质，但某些奇异的行为被它们排除在外。为了更好地理解这两种正则性及其相互关系，需要现有技术的发展，这将有利于C*-代数分类的一般程序。所选导师与研究人员的兴趣相匹配，并在研究领域发表了多篇论文。他的出版物包括将Cuntz半群视为附加不变量的理由，这与Elliott的分类原则相协调。此外，他的出版物为研究项目提供了有前途的工具。

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.