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

http://dx.doi.org/10.4153/CJM-2005-016-5
Canad. J. Math. 57(2005), 351-399

  Published:2005-04-01
 Printed: Apr 2005

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.