Stability of $C^{\ast }$-algebras is not a stable property

Abstract

We show that there exists a $C^{\ast }$-algebra $B$ such that $M_2(B)$ is stable, but $B$ is not stable. Hence stability of $C^{\ast }$-algebras is not a stable property. More generally, we find for each integer $n\ge 2$ a $C^{\ast }$-algebra $B$ so that $M_n(B)$ is stable and $M_k(B)$ is not stable when $1\le k<n$. The $C^{\ast }$-algebras we exhibit have the additional properties that they are simple, nuclear and of stable rank one.