  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 $\alpha=\omega_1$.
