- Baum–Connes conjecture
In
mathematics , specifically inoperator K-theory , the Baum–Connes conjecture suggests a link between theC*-algebra of a group and theK-homology of the correspondingclassifying space of proper action s of that group.It thus sets up a correspondence between different areas of mathematics, the K-homology being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the reduced -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 theNovikov conjecture .The conjecture is also closely related to
index theory , as the assembly map is a sort of index, and it plays a major role inAlain Connes 'snoncommutative geometry program.The origins of the conjecture go back to
Fredholm theory , theAtiyah-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.Formulation
Let Γ be a second countable
locally compact group (for instance a countablediscrete group ). One can define a morphism:called the assembly map, from theequivariant K-homology with -compact supports of theclassifying space of proper action s to the K-theory of thereduced C*-algebra of Γ. The index * can be 0 or 1.Paul Baum andAlain Connes introduced the following conjecture (1982) about this morphism::The assembly map μ is an
isomorphism .As the left hand side tends to be more easily accessible than the right hand side, because there are hardly any general structure theorems of the -algebra, one usually views the conjecture as an "explanation" of the right hand side.
The original formulation of the conjecture was somewhat different, as the notion of equivariant K-homology was not yet common in 1982.
In case is discrete and torsion-free, the left hand side reduces to the non-equivariant K-homology with compact supports of the ordinary classifying space of .
There is also more general form of the conjecture, known as Baum-Connes conjecture with coefficients, where both sides have coefficients in the form of a -algebra on which acts by -automorphisms. It says in KK-language that the assembly
is an isomorphism, containing the case without coefficients as the case .However, counterexamples to the conjecture with coefficients were found in 2002 by
Nigel Higson ,Vincent Lafforgue andGeorge Skandalis , basing on not universally accepted, as of 2008, results of Gromov on expanders in Cayley graphs. Even provided validity of Higson, Lafforgue & Skandalis, conjecture with coefficients remains an active area of research, since it is, not unlike the classical conjecture, often seen as a statement concerning particular groups or class of groups.Example
Let be the integers . Then the left hand side is the
K-homology of which is the circle. The -algebra of the integers is by the commutative Gelfand-Naimark-transform, which reduces to theFourier transform in this case, isomorphic to the algebra of continuous functions on the circle. So the right hand side is the topological K-theory of the circle. One can then show that the assembly map is KK-theoreticPoincaré duality as defined byGuennadi Kasparov , which is an isomorphism.Results
The conjecture without coefficients is still open, although the field has received great attention since 1982.The conjecture is proved for the following classes of groups:
* Discrete subgroups of and .
* Groups with theHaagerup property , sometimes called a-T-menable groups. These are groups that admit an isometric action on an affine Hilbert space which is proper in the sense that for all and all sequences of group elements with . Examples of a-T-menable groups are amenable groups,Coxeter group s, groups acting properly on trees, and groups acting properly on simply connected cubical complexes.
* Groups that admit a finite presentation with only one relation.
* Discrete cocompact subgroups of real Lie groups of real rank 1.
* Cocompact lattices in , or . It was a long-standing problem since the first days of the conjecture to expose a single infinite property T-group that satisfies it. However, such a group was given by V. Lafforgue in 1998 as he showed that cocompact lattices in have the property of rapid decay and thus satisfy the conjecture.
* Gromov hyperbolic groups and their subgroups.
* Among non-discrete groups, the conjecture has been shown in 2003 by J. Chabert, S. Echterhoff and R. Nest for the vast class of all almost connected groups (i. e. groups having a cocompact connected component), and all groups of -rational points of a linear algebraic group over alocal field of characteristic zero (e.g. ). For the important subclass of real reductive groups, the conjecture had already been shown in 1982 by A. Wassermann.Injectivity is known for a much larger class of groups thanks to the Dirac-dual-Dirac method. This goes back to ideas ofMichael Atiyah and was developed in great generality byGennadi Kasparov in 1987.Injectivity is known for the following classes:
* Discrete subgroups of connected Lie groups or virtually connected Lie groups.
* Discrete subgroups of p-adic groups.
* Bolic groups (a certain generalization of hyperbolic groups).
* Groups which admit an amenable action on some compact space.The simplest example of a group for which it is not known whether it satisfies the conjecture is .
References
*Guido Mislin and Alain Valette (2003), "Proper Group Actions and the Baum-Connes Conjecture" ISBN 978-3-7643-0408-9
External links
* [http://www.math.ist.utl.pt/~matsnev/BCexpository.pdf On the Baum-Connes conjecture] by Dmitry Matsnev.
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