Corona algebras, and quantum groups

点击翻译按钮获取中文摘要

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.