Shapiro's lemma for topological K-theory of groups

Abstract

Abstract. Let X G be the crossed product groupoid of a locally compact group G acting on a locally compact space X. For any X G-algebra A we show that a natural forgetful map from the topological K-theory K top * (X G; A) of the groupoid X G with coefficients in A to the topological K-theory K top * (G; A) of G with coefficients in A is an isomorphism. We then discuss several interesting consequences of this result for the Baum-Connes conjecture.

Equivariant E-theory for groupoids acting on C∗-algebras

“…Thus, we can revisit some construction from [18] in this setting (cf. [21]) and also obtain some results about groups by considering groupoids [6].…”

“…It is noteworthy that the coarse version of the Baum-Connes conjecture can be seen as a particular case of the groupoid formulation [32]. Moreover, the groupoid case turned out to be a source of inspiration for the group case [6].…”

Going-down functors, the K�nneth formula, and the Baum-Connes conjecture

“…Since the action of G on X is amenable, it follows from [Tu] (but see also [CEO,Corollary 0.4]) that the bottom horizontal arrow is an isomorphism. Since X is convex and the action of G on X is affine, the space X is K-equivariantly contractible for any compact subgroup K of G. An easy diagram chase then shows the split-injectivity of the assembly map µ A , provided that we have the following extension of [H,Proposition 3.7] to arbitrary second countable locally compact groups.…”

“…Using our general method (together with [CEO,Corollary 0.4]) we can extend their arguments to obtain Theorem 1.11.…”

“…Since A ⊗ D is an X G-algebra and the action of G on X is topologically amenable, it follows from [CEO,Corollary 0.4] that the middle vertical arrow is an isomorphism. An easy diagram chase then gives the result.…”