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Classification of Simple Tracially AF C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Huaxin Lin
Huaxin Lin*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai, China
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Abstract

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We prove that pre-classifiable (see 3.1) simple nuclear tracially $\text{AF}\,\,{{C}^{*}}$-algebras $\left( \text{TAF} \right)$ are classified by their $K$-theory. As a consequence all simple, locally $\text{AH}$ and $\text{TAF}\,\,\,{{C}^{*}}$-algebras are in fact $\text{AH}$ algebras (it is known that there are locally $\text{AH}$ algebras that are not $\text{AH}$). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple $\text{TAF}\,\,\,{{C}^{*}}$-algebras with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$-theory and ${{K}_{0}}{{\left( A \right)}_{+}}$ is locally finitely generated, then $A\,\otimes \,Q\,\cong \,B\,\otimes \,Q$, where $Q$ is the $\text{UHF}$-algebra with the rational ${{K}_{0}}$. Classification results (with restriction on ${{K}_{0}}$ - theory) for the above ${{C}^{*}}$-algebras are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple $\text{TAF}\,\,\,{{C}^{*}}$-algebras with the unique normalized trace satisfying the $\text{UCT}$ and with ${{K}_{1}}\left( A \right)\,=\,{{K}_{1}}\left( B \right)$, and $A$ and $B$ have the same rational (scaled ordered) ${{K}_{0}}$, then $A\,\cong \,B$. Similar results are also obtained for some cases in which ${{K}_{0}}$ is non-divisible such as ${{K}_{0}}\left( A \right)\,=\,\mathbf{Z}\left[ 1/2 \right]$.

Keywords

46L0546L35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 1 , 01 February 2001 , pp. 161 - 194
Copyright
Copyright © Canadian Mathematical Society 2001

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