nLab
assembly map

Contents

Contents

Idea

The analytic assembly map is a natural morphism from GG-equivariant topological K-theory to the operator K-theory of a corresponding crossed product C*-algebra.

More generally in equivariant KK-theory this is called the Kasparov descent map and is of the form

KK G(A,B)KK(GA,GB) KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)

where on the left we have GG-equivariant KK-theory and on the right ordinary KK-theory of crossed product C*-algebras (which by the discussion there are models for the groupoid convolution algebras of GG-action groupoids).

(recalled as Blackadar, theorem 20.6.2)

Properties

The Baum-Connes conjecture states that under some conditions the analytic assembly map is in fact an isomorphism. The Novikov conjecture makes statements about it being an injection. The Green-Julg theorem states that under some (milder) conditions the Kasparov desent map is an isomorphism.

References

The construction goes back to

  • Gennady Kasparov, The index of invariant elliptic operators, K-theory, and Lie group representations. Dokl. Akad. Nauk. USSR, vol. 268, (1983), 533-537.

An introduction is in

A textbook account is in

See also

  • Markus Land, The Analytical Assembly Map and Index Theory, (arXiv:1306.5657)

Last revised on July 10, 2013 at 01:06:11. See the history of this page for a list of all contributions to it.