Stability of $C^*$-Algebras is Not a Stable Property
We show that there exists a $C^*$-algebra $B$ such that $M_2(B)$ is stable, but $B$ is not stable. Hence stability of $C^*$-algebras is not a stable property. More generally, we find for each integer $n \ge 2$ a $C^*$-algebra $B$ so that $M_n(B)$ is stable and $M_k(B)$ is not stable when $1 \le k < n$. The $C^*$-algebras we exhibit have the additional properties that they are simple, nuclear and of stable rank one.
The construction is similar to Jesper Villadsen's construction in  of a simple $C^*$-algebra with perforation in its ordered $K_0$-group.
1991 Mathematics Subject Classification: 46L05, 46L35, 19K14
Keywords: Stable $C^*$-algebras, perforation in $K_0$, scaled ordered Abelian groups.
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