Corona algebras, and quantum groups

C*-代数是拓扑空间上函数代数的非交换推广，因此有时也称为非交换拓扑空间。用于研究C*-代数拓扑的方法，也被称为几何泛函分析，与经典拓扑学的方法有很大的不同，但是对所有C*-代数进行分类的雄心勃勃的计划就像通过拓扑不变量对拓扑空间进行分类的经典计划一样。最近的反例表明，可能需要新的方法。然而，现有理论的一个亮点是基希贝格对纯无限单C*-代数的分类，通过推广基希贝格的一个关键定理，我们得到了冠因子分解性质，它以解析的形式表征了非交换拓扑性质，并对C*-代数的行为产生了深远的影响。

A C*-algebra is a noncommutative generalization of a function algebra over a topological space, thus is sometimes called a noncommutative topological space. The methods used for studying the topology of C*-algebras, also known as geometric functional analysis, are very different from those of classical topology, but the rather ambitious program of classifying all C*-algebras is like the classical program of classifying topological spaces by means of their topological invariants. The recent plethoria of counterexamples indicate that new approaches may be needed. However, one of the high points of the existing theory is Kirchberg's classification of purely infinite simple C*-algebras, and by generalizing one of Kirchberg's key theorems, we obtain the corona factorization property, which characterizes a noncommutative topology property in analytic terms and has deep consequences for the behaviour of a C*-algebra.