Stable rank of inclusion of C*-algebras of depth 2

Abstract

Let $1 \in A \subset B$ be an inclusion of unital C*-algebras of index-finite type and depth 2. Suppose that $A$ is infinite dimensional simple with $tsr(A) = 1$ and SP-property. Then $tsr(B) \leq 2$. As a corollary when $A$ is a simple C*-algebra with $\tsr(A) = 1$ and SP-property and $\alpha$ an action of a finite group $G$ on $\Aut(A)$, $\tsr(A \rtimes_\alpha G) \leq 2$.