The Baum-Connes conjecture, in operator K-theory, suggests a link between the K-theory of a group's reduced C*-algebra and the K-homology of the classifying space of its proper actions, setting up a correspondence between analytic and geometric areas of mathematics. If true, the conjecture would imply older results as consequences: its surjectivity part implies the Kadison-Kaplansky conjecture for discrete torsion-free groups, and its injectivity part is closely related to the Novikov conjecture. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe Baum-Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group, via an assembly map. 1 Proposed Year Progress Toward ResolutionThe conjecture without coefficients remains open, though counterexamples to the more general conjecture with coefficients were found in 2002 by Higson, Lafforgue and Skandalis. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Baum-Connes conjecture (Wikipedia)
Posed By
Source Baum-Connes conjecture (Wikipedia)
Sources
1. Baum-Connes Conjecture (Wikipedia)
Wikimedia FoundationLead section
the Baum-Connes conjecture suggests a link between the K-theory of the
Lead section
In mathematics, specifically in operator K-theory, the Baum-Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group.
Formulation section
Paul Baum and Alain Connes introduced the following conjecture (1982) about this morphism
Results section, 2002 counterexample sentence
counterexamples to the conjecture with coefficients were found in 2002 by Nigel Higson, Vincent Lafforgue and Georges Skandalis.
Results section, open-without-coefficients sentence
The conjecture without coefficients is still open, although the field has received great attention since 1982.
View the Source Baum-Connes conjecture (Wikipedia)
In Branch: Analysis, Lead
In mathematics, specifically in operator K-theory, the Baum-Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group.
Posed By: Alain Connes, History
Paul Baum and Alain Connes introduced the following conjecture (1982) about this morphism
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.