The Baum-Connes Conjecture for Groupoids

Abstract

Given a (not necessarily discrete) proper metric space M with bounded geometry, we define a groupoid G(M ). We show that the coarse Baum-Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M ) is an isomorphism, is hereditary by taking closed subspaces.

Virtual groups 45 years later

“…Jolissaint [Jo05] has done exactly this at least for measure preserving relations. In fact a little earlier Tu [Tu00] had defined exactly such a property for topological groupoids (also see below in section 10). One could perhaps pick nits and argue that in the title of Tu's paper one should substitute "a-T -moyennables" for "moyennables".…”

“…The connected Lie groups with Haagerup property can be completely classified-[CCJJV01, p. 41]. There is a related property called K-amenability introduced by Cuntz [Cun83], which is now known to be implied by the Haagerup Property [HiKa97], [HiKa01], and also [Tu99b] and [Tu00]. There are groups that are K-amenable but do not have the Haagerup property, for example the semi-direct product of Z 2 with SL 2 (Z)-[CCJJV01, p. 8].…”

“…Again, following the lead from group theory, one can introduce the notion of a topological groupoid having the Haagerup property or being a-T -menable. The definition is implicit in [Tu99a,Tu99b] and more explicit in [Tu00] in which certain properties of the C * -algebra of the groupoid are established that are parallel to properties of the C * -algebra of an group with the Haagerup property.…”

“…The conjecture was originally stated for discrete groups but its domain was extended to include continuous groups, foliation groupoids, and then locally compact groupoids [Tu00]. So let G be a locally compact Hausdorff groupoid with a continuous Haar system and C * r (G) be its reduced C * -algebra.…”

The ring structure for equivariant twisted K-theory

“…We also recall [4,41] that if A is a Γ − C * -algebra, there is a group morphism, the Baum-Connes assembly map…”