eBook ISBN:  9781470422257 
Product Code:  MEMO/235/1107.E 
List Price:  $80.00 
MAA Member Price:  $72.00 
AMS Member Price:  $48.00 
eBook ISBN:  9781470422257 
Product Code:  MEMO/235/1107.E 
List Price:  $80.00 
MAA Member Price:  $72.00 
AMS Member Price:  $48.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 235; 2015; 109 pp
A unital separable \(C^\ast\)algebra, \(A\) is said to be locally AH with no dimension growth if there is an integer \(d>0\) satisfying the following: for any \(\epsilon >0\) and any compact subset \({\mathcal F}\subset A,\) there is a unital \(C^\ast\)subalgebra, \(B\) of \(A\) with the form \(PC(X, M_n)P\), where \(X\) is a compact metric space with covering dimension no more than \(d\) and \(P\in C(X, M_n)\) is a projection, such that \( \mathrm{dist}(a, B)<\epsilon \text{ for all } a\in\mathcal {F}.\)
The authors prove that the class of unital separable simple \(C^\ast\)algebras which are locally AH with no dimension growth can be classified up to isomorphism by their Elliott invariant. As a consequence unital separable simple \(C^\ast\)algebras which are locally AH with no dimension growth are isomorphic to a unital simple AHalgebra with no dimension growth.

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. Definition of ${\mathcal C}_g$

4. $C^*$algebras in ${\mathcal C}_g$

5. Regularity of $C^*$algebras in ${\mathcal C}_1$

6. Traces

7. The unitary group

8. ${\mathcal Z}$stability

9. General Existence Theorems

10. The uniqueness statement and the existence theorem for Bott map

11. The Basic Homotopy Lemma

12. The proof of the uniqueness theorem 10.4

13. The reduction

14. Appendix


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A unital separable \(C^\ast\)algebra, \(A\) is said to be locally AH with no dimension growth if there is an integer \(d>0\) satisfying the following: for any \(\epsilon >0\) and any compact subset \({\mathcal F}\subset A,\) there is a unital \(C^\ast\)subalgebra, \(B\) of \(A\) with the form \(PC(X, M_n)P\), where \(X\) is a compact metric space with covering dimension no more than \(d\) and \(P\in C(X, M_n)\) is a projection, such that \( \mathrm{dist}(a, B)<\epsilon \text{ for all } a\in\mathcal {F}.\)
The authors prove that the class of unital separable simple \(C^\ast\)algebras which are locally AH with no dimension growth can be classified up to isomorphism by their Elliott invariant. As a consequence unital separable simple \(C^\ast\)algebras which are locally AH with no dimension growth are isomorphic to a unital simple AHalgebra with no dimension growth.

Chapters

1. Introduction

2. Preliminaries

3. Definition of ${\mathcal C}_g$

4. $C^*$algebras in ${\mathcal C}_g$

5. Regularity of $C^*$algebras in ${\mathcal C}_1$

6. Traces

7. The unitary group

8. ${\mathcal Z}$stability

9. General Existence Theorems

10. The uniqueness statement and the existence theorem for Bott map

11. The Basic Homotopy Lemma

12. The proof of the uniqueness theorem 10.4

13. The reduction

14. Appendix