assembly map

noncommutative topology, noncommutative geometry

noncommutative stable homotopy theory

**genus, orientation in generalized cohomology**

The *analytic assembly map* is a natural morphism from $G$-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

$KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)$

where on the left we have $G$-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 $G$-action groupoids).

(recalled as Blackadar, theorem 20.6.2)

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.

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

- Alain Valette,
*Introduction to the Baum-Connes conjecture*(pdf)

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.