eBook ISBN:  9781470402945 
Product Code:  MEMO/148/703.E 
List Price:  $51.00 
MAA Member Price:  $45.90 
AMS Member Price:  $30.60 
eBook ISBN:  9781470402945 
Product Code:  MEMO/148/703.E 
List Price:  $51.00 
MAA Member Price:  $45.90 
AMS Member Price:  $30.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 148; 2000; 86 ppMSC: Primary 19; 46; Secondary 22
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.
ReadershipGraduate students and research mathematicians interested in operator algebras and noncommutative geometry, specifically the BaumCannes and Novikov conjectures.

Table of Contents

Chapters

Introduction

1. Asymptotic morphisms

2. The homotopy category of asymptotic morphisms

3. Functors on the homotopy category

4. Tensor products and descent

5. $C$*algebra extensions

6. Etheory

7. Cohomological properties

8. Proper algebras

9. Stabilization

10. Assembly

11. The Green–Julg theorem

12. Induction and compression

13. A generalized Green–Julg theorem

14. Application to the Baum–Connes conjecture

15. Concluding remark on assembly for proper algebras


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

Chapters

Introduction

1. Asymptotic morphisms

2. The homotopy category of asymptotic morphisms

3. Functors on the homotopy category

4. Tensor products and descent

5. $C$*algebra extensions

6. Etheory

7. Cohomological properties

8. Proper algebras

9. Stabilization

10. Assembly

11. The Green–Julg theorem

12. Induction and compression

13. A generalized Green–Julg theorem

14. Application to the Baum–Connes conjecture

15. Concluding remark on assembly for proper algebras