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

Huaxin Lin
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Abstract

Let $A$ be an amenable separable \CA and $B$ be a non-unital but $σ$-unital simple \CA with continuous scale. We show that two essential extensions $τ_1$ and $τ_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$ [τ_1]=[τ_2] {\rm in} KL(A, M(B)/B). $$ 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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Huaxin Lin. 2004-01-19. Extensions by simple $C^*$-algebras -- Quasidiagonal extensions. https://arxiv.org/abs/math/0401241

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