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.
Introduction Asymptotic morphisms The homotopy category of asymptotic morphisms Functors on the homotopy category Tensor products and descent $C^\ast$-algebra extensions $E$-theory Cohomological properties Proper algebras Stabilization Assembly The Green-Julg theorem Induction and compression A generalized Green-Julg theorem Application to the Baum-Connes conjecture Concluding remarks on assembly for proper algebras References.