Topological stable rank of inclusions of unital C*-algebras

Abstract

Let $1 \in A \subset B$ be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute topological stable rank of $B$ ($= \tsr(B)$) when $A$ has topological stable rank one. We show that $\tsr(B) \leq 2$ when $A$ is a tsr boundedly divisible algebra, in particular, $A$ is a C*-minimal tensor product $UHF \otimes D$ with $\tsr(D) = 1$. When $G$ is a finite group and $\alpha$ is an action of $G$ on UHF, we know that a crossed product algebra $UHF \rtimes_\alpha G$ has topological stable rank less than or equal to two. These results are affirmative datum to a generalization of a question by B. Blackadar in 1988.