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。 群是编码结构的对称性的数学对象，并且可以用于构造群代数。量子群被认为是群的推广，这与C*-代数被认为是非交换拓扑空间的意义非常相似。C*-代数也可能是量子群。我们将C*-代数分类程序的强大机制应用于某类量子群，获得了有趣的结果，可能最终导致我们得到量子群的新例子。这是一种非常新颖的方法来解决量子群分类的挑战性问题。

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. Groups are mathematical objects that encode symmetry properties of a structure, and can be used to construct group algebras. Quantum groups are thought of as generalizations of groups, much in the same sense as C*-algebras are thought of as noncommutative topological spaces. It is possible for a C*-algebra to also be a quantum group. We apply the powerful machinery of the C*-algebraic classification program to a certain class of quantum group, obtaining interesting results, possibly eventually leading us to new examples of quantum groups. This is a highly novel approach to the challenging problem of classification of quantum groups.