A non-stable C*-algebra with an elementary essential composition series

A C*-algebra $A$ is said to be stable if it is isomorphic to $A \otimes K(\ell_2)$. Hjelmborg and R\o rdam 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 $A$ of $B(\ell_2)$, which is the inductive limit of length $\omega_1$ of its separable stable ideals $I_\alpha$ ($\alpha<\omega_1$) satisfying $I_{\alpha+1}/I_\alpha\cong K(\ell_2)$ for each $\alpha<\omega_1$, while $A$ is not stable. The sequence $(I_\alpha)_{\alpha\leq\omega_1}$ is the GCR composition series of $A$ which in this case coincides with the Cantor-Bendixson composition series as a scattered C*-algebra. $A$ has the property that all of its proper two-sided ideals are listed as $I_\alpha$s for some $\alpha<\omega_1$ and therefore the family of stable ideals of $A$ has no maximal element. By taking $A'=A\otimes K(\ell_2)$ we obtain a stable C*-algebra with analogous composition series $(J_\alpha)_{\alpha<\omega_1}$ whose ideals $J_\alpha$s are isomorphic to $I_\alpha$s for each $\alpha<\omega_1$. In particular, there are nonisomorphic scattered C*-algebras whose GCR composition series $(I_\alpha)_{\alpha\leq\omega_1}$ satisfy $I_{\alpha+1}/I_\alpha\cong K(\ell_2)$ for all $\alpha<\omega_1$, for which the composition series differ first at $\alpha=\omega_1$.