Bivariant KK-Theory and the Baum–Connes conjecure
The extension of K-theory from topological spaces to operator algebras provides the most powerful tool for the study of C *-algebras. On one side there now exist far reaching classification results in which certain classes of C *-algebras can be classified by their K-theoretic data. This started with the early work of Elliott [Ell76] on the classification of AF-algebras – inductive limits of finite-dimensional C *-algebras. It went on with the classification of simple, separable, nuclear, purely infinite C *-algebras by Kirchberg and Phillips [KP00,Phi00]. Presently, due to the work of many authors (e.g., see [Win16] for a survey on the most recent developments) the classification program covers a very large class of nuclear algebras.
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