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.

Readership

Graduate students and research mathematicians interested in operator algebras and noncommutative geometry, specifically the Baum-Cannes and Novikov conjectures.

Table of Contents

• 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