# Locally AH-Algebras

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*Huaxin Lin*

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 AH-algebra with no dimension growth.

#### Table of Contents

# Table of Contents

## Locally AH-Algebras

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Chapter 2. Preliminaries 512 free
- Chapter 3. Definition of 𝒞_{ℊ} 1320
- Chapter 4. \CA s in 𝒞_{ℊ} 1522
- Chapter 5. Regularity of \CA s in \cal𝐶₁ 2330
- Chapter 6. Traces 3542
- Chapter 7. The unitary group 4148
- Chapter 8. \cal𝑍-stability 4956
- Chapter 9. General Existence Theorems 5360
- Chapter 10. The uniqueness statement and the existence theorem for Bott map 6774
- Chapter 11. The Basic Homotopy Lemma 7582
- Chapter 12. The proof of the uniqueness theorem 10.4 8996
- Chapter 13. The reduction 97104
- Chapter 14. Appendix 103110
- Bibliography 107114
- Back Cover Back Cover1122