Memoirs of the American Mathematical Society 2000; 86 pp; softcover Volume: 148 ISBN10: 0821821164 ISBN13: 9780821821169 List Price: US$48 Individual Members: US$28.80 Institutional Members: US$38.40 Order Code: MEMO/148/703
 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 sixterm 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 BaumConnes 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 BaumCannes 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 GreenJulg theorem
 Induction and compression
 A generalized GreenJulg theorem
 Application to the BaumConnes conjecture
 Concluding remarks on assembly for proper algebras
 References
