A nonstable $C^*$-algebra with an elementary essential composition series

Abstract:

A $C^*$-algebra $\mathcal {A}$ is said to be stable if it is isomorphic to $\mathcal {A} \otimes \mathcal {K}(\ell _2)$. Hjelmborg and Rørdam have shown that countable inductive limits of separable stable $C^*$-algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra $\mathcal {A}$ of $\mathcal {B}(\ell _2)$, which is the inductive limit of length $\omega _1$ of its separable stable ideals $\mathcal {I}_\alpha$ ($\alpha <\omega _1$) satisfying $\mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2)$ for each $\alpha <\omega _1$, while $\mathcal {A}$ is not stable. The sequence $(\mathcal {I}_\alpha )_{\alpha \leq \omega _1}$ is the GCR composition series of $\mathcal {A}$ which in this case coincides with the Cantor–Bendixson composition series as a scattered $C^*$-algebra. $\mathcal {A}$ has the property that all of its proper two-sided ideals are listed as $\mathcal {I}_\alpha$’s for some $\alpha <\omega _1$, and therefore the family of stable ideals of $\mathcal {A}$ has no maximal element.

By taking $\mathcal {A}’=\mathcal {A}\otimes \mathcal {K}(\ell _2)$ we obtain a stable $C^*$-algebra with analogous composition series $(\mathcal {J}_\alpha )_{\alpha <\omega _1}$ whose ideals $\mathcal {J}_\alpha$ are isomorphic to $\mathcal {I}_\alpha$ for each $\alpha <\omega _1$. In particular, there are nonisomorphic scattered $C^*$-algebras whose GCR composition series $(\mathcal {I}_\alpha )_{\alpha \leq \omega _1}$ satisfy $\mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2)$ for all $\alpha <\omega _1$, for which the composition series differs first at $\alpha =\omega _1$.