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Extensions by Simple $C^*$-Algebras: Quasidiagonal Extensions

Published:2005-04-01
Printed: Apr 2005
• Huaxin Lin
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Abstract

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 $$[\tau_1]=[\tau_2] \text{ in } KL(A, M(B)/B).$$ 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 $KL(A, M(B)/B)$. Using $KL(A, M(B)/B)$, 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: Extensions, Simple $C^*$-algebras
 MSC Classifications: 46L05 - General theory of $C^*$-algebras 46L35 - Classifications of $C^*$-algebras