Extensions by simple $C^*$-algebras -- Quasidiagonal extensions

Details Extensions by simple $C^*$-algebras -- Quasidiagonal extensions Extensions by simple $C^*$-algebras -- Quasidiagonal extensions

2004-01-19

arxiv.org/abs/math/0401241v1

Mathematics

Operator Algebras

to appear Canad J. Math

Scientific paper

Let $A$ be an amenable separable \CA and $B$ be a non-unital but $\sigma$-unital simple \CA 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 $$ [\tau_1]=[\tau_2] {\rm in} KL(A, M(B)/B). $$ 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.

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