BaumConnes

BaumConnes is a conjecture in mathematics, specifically in the field of operator algebras and noncommutative geometry. It was proposed by Alain Connes in 1982 and is named after the mathematician Adolphe Baumé, who was a contemporary of Connes. The conjecture relates the K-theory of a discrete group to the K-theory of its reduced group C*-algebra. This is a significant result in the study of operator algebras and has implications for the classification of C*-algebras.

The BaumConnes conjecture states that for a locally compact group G, there is a natural isomorphism between the K-theory of the reduced group C*-algebra of G and the K-theory of the space of proper G-equivariant vector bundles over a classifying space for G. This conjecture has been proven for many classes of groups, including amenable groups, groups of type I, and groups with the Haagerup property. However, the full conjecture remains open.

The BaumConnes conjecture is important because it provides a way to compute the K-theory of a group using geometric data. This has applications in various areas of mathematics, including topology, geometry, and number theory. It also has implications for the classification of C*-algebras, which are a fundamental object of study in operator algebras. The conjecture is a central topic in the study of operator algebras and noncommutative geometry, and its proof or disproof would have significant implications for the field.