Extensions of Stable $C^*$-Algebras

We show that an extension of two stable $C^*$-algebras need not be stable. More explicitly we find an extension $$0 \to C(Z) \otimes {\cal K} \to A \to {\cal K} \to 0$$ for some (infinite dimensional) compact Hausdorff space $Z$ such that $A$ is not stable. The $C^*$-algebra $A$ in our example has an approximate unit consisting of projections.

2000 Mathematics Subject Classification: 46L05, 46L35

Keywords and Phrases: stable $C^*$-algebras, extensions

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