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 [7] 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.

Full text: dvi.gz 23 k, dvi 59 k, ps.gz 83 k, pdf 172 k

Home Page of DOCUMENTA MATHEMATICA