Extensions by Simple C*-Algebras: Quasidiagonal Extensions

Lin, Huaxin

Canadian journal of mathematics, Tome 57 (2005) no. 2, pp. 351-399

Résumé

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.

Détail

DOI : 10.4153/CJM-2005-016-5

Mots-clés : 46L05, 46L35

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