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Equivariant $$E$$-Theory for $$C^*$$-Algebras
Erik Guentner, Indiana University-Purdue University Indianapolis, IN, Nigel Higson, Pennsylvania State University, University Park, and Jody Trout, Dartmouth College, Hanover, NH
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Memoirs of the American Mathematical Society
2000; 86 pp; softcover
Volume: 148
ISBN-10: 0-8218-2116-4
ISBN-13: 978-0-8218-2116-9
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 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.

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

• 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