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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
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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
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