# Equivariant \(E\)-Theory for \(C^*\)-Algebras

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*Erik Guentner; Nigel Higson; Jody Trout*

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.

#### Table of Contents

# Table of Contents

## Equivariant $E$-Theory for $C^{*}$-Algebras

- Contents vii8 free
- Introduction 110 free
- Chapter 1. Asymptotic Morphisms 413 free
- Chapter 2. The Homotopy Category of Asymptotic Morphisms 817
- Chapter 3. Functors on the Homotopy Category 1928
- Chapter 4. Tensor Products and Descent 2332
- Chapter 5. C*-Algebra Extensions 2938
- Chapter 6. E-Theory 4049
- Chapter 7. Cohomological Properties 4958
- Chapter 8. Proper Algebras 5362
- Chapter 9. Stabilization 5564
- Chapter 10. Assembly 5968
- Chapter 11. The Green–Julg Theorem 6372
- Chapter 12. Induction and Compression 6776
- Chapter 13. A Generalized Green–Julg Theorem 7685
- Chapter 14. Application to the Baum–Connes Conjecture 7988
- Chapter 15. Concluding Remarks on Assembly for Proper Algebras 8392
- References 8594