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Extensions by Simple C*-Algebras: Quasidiagonal Extensions

Published online by Cambridge University Press:  20 November 2018

Huaxin Lin
Huaxin Lin*
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, U.S.A. and Department of Mathematics, East China Normal University, Shanghai, China
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Abstract

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Let $A$ be an amenable separable ${{C}^{*}}$-algebra and $B$ be a non-unital but $\sigma $-unital simple ${{C}^{*}}$- algebra with continuous scale. We show that two essential extensions ${{\tau }_{1}}$ and ${{\tau }_{2}}$ of $A$ by $B$ are approximately unitarily equivalent if and only if

$$\left[ {{\tau }_{1}} \right]\,=\,\left[ {{\tau }_{2}} \right]\,\text{in}\,KL\left( A,\,M\left( B \right)/B \right).$$

If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL\left( A,\,M\left( B \right)/B \right)$. Using $KL\left( A,\,M\left( B \right)/B \right)$, we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.

Keywords

46L0546L35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 2 , 01 April 2005 , pp. 351 - 399
Copyright
Copyright © Canadian Mathematical Society 2005

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