Mikael Rørdam

摘要

We show that there exists a $C^*$-algebra $B$ such that $M_2(B)$ isstable, but $B$ is not stable. Hence stability of $C^*$-algebras isnot a stable property. More generally, we find for each integer $n ge2$ a $C^*$-algebra $B$ so that $M_n(B)$ is stable and $M_k(B)$ is notstable when $1 le k n$. The $C^*$-algebras we exhibit have theadditional properties that they are simple, nuclear and of stable rankone.