Baum–Connes conjecture

In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the C*-algebra of a group and the K-homology of the corresponding classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the reduced {\displaystyle C^{*}}C^*-algebra is a purely analytical object.

The conjecture, if true, would have some older famous conjectures as consequences. For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for a discrete torsion-free group, and the injectivity is closely related to the Novikov conjecture.

The conjecture is also closely related to index theory, as the assembly map {\displaystyle \mu }\mu is a sort of index, and it plays a major role in Alain Connesnoncommutative geometry program.

The origins of the conjecture go back to Fredholm theory, the Atiyah–Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects.

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  1.  Introduction to the Baum-Connes Conjecture Alain VALETTE (PDF)
  2. A Survey on the Baum-Connes Conjecture (PDF)