Equivariant E-theory for C -algebras

Let $A$ and $B$ be $C^*$-algebras which are equipped with continuous actions of a second countable, locally compact group $G$. We define a notion of equivariant asymptotic morphism, and use it to define equivariant $E$-theory groups $E_G(A,B)$ which generalize the $E$-theory groups of Connes and Higson. We develop the basic properties of equivariant $E$-theory, including a composition product and six-term exact sequences in both variables, and apply our theory to the problem of calculating $K$-theory for group $C^*$-algebras. Our main theorem gives a simple criterion for the assembly map of Baum and Connes to be an isomorphism. The result plays an important role in recent work of Higson and Kasparov on the Baum-Connes conjecture for groups which act isometrically and metrically properly on Hilbert space.