nLab assembly map

Idea

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)

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.

Related concepts

• Novikov conjecture

• Baum-Connes conjecture

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

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

A textbook account is in

• Bruce Blackadar, K-Theory for Operator Algebras

See also

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