We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

Nous construisons une $KK$-théorie pour les algèbres de Banach équivariante par l’action d’un groupoïde et nous montrons la conjecture de Baum–Connes à coefficients commutatifs pour les groupes hyperboliques et pour les groupoïdes de Poincaré des feuilletages à base compacte qui peuvent être munis d’une métrique riemannienne longitudinale à courbure sectionnelle strictement négative.

We construct a $KK$-theory for Banach algebras, equivariant with respect to the action of a groupoid. We prove the Baum–Connes conjecture with commutative coefficients for hyperbolic groups and for the Poincaré groupoids of foliations with a compact base and a longitudinal Riemannian metrics with negative sectional curvature.

We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphism and thus to the Dirac dual Dirac method of attacking the Novikov conjecture.