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

C*-代数分类中的电晕分解性质和Cuntz半群

基本信息

批准号：
236576044
负责人：
Henning Petzka
金额：
--
依托单位：
Mathematisches Institut
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2013
资助国家：
德国
起止时间：
2012-12-31 至 2013-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/236576044?language=en
关键词：
corona factorization property Cuntz semigroup

项目摘要

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.

1989年，乔治埃利奥特提出可分顺从C*-代数可以完全分类的K-理论和迹。虽然在接下来的几年里验证非常成功，但最终发现了反例。因此，对于一个完整的分类，不变量的扩展是必要的。Cuntz半群是扩展不变量的一个候选者，但在分类程序中使用它仍然具有挑战性。我的目标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。我的论文的结果产生于试图理解（不可分类的）C*-代数的两个特定正则性性质的相关性。这两个性质适用于所有的C*-代数，对于这些C*-代数，可以用原始的艾略特不变量进行分类。艾略特猜想的反例也可以有这两个性质，但它们排除了某些奇异的行为。为了更好地理解这两个正则性质及其相关性，需要发展现有的技术，这将有利于C*-代数分类的一般程序。所选的导师符合研究人员的兴趣，并在研究领域有几个出版物。他的出版物包括理由考虑Cuntz半群作为一个额外的不变量，这调和的原则艾略特的分类。他的出版物也为研究项目提供了有前途的工具。