article

Title: Cancellation for inclusions of C*-algebras of finite depth

Authors: Ja A Jeong, N. Christopher Phillips, Hiroyuki Osaka and Tamotsu Teruya

Issue: Volume 58 (2009), Issue 4, 1537-1564

Abstract:

$Let $1 \in A \subset B$ be a pair of $C^{*}$-algebras with common unit. We prove that if $E\colon B \to A$ is a conditional expectation with index-finite type and a quasi-basis of $n$ elements, then the topological stable rank satisfies \[ \mathop{tsr}(B) \leq \mathop{tsr}(A)+n-1. \] As an application we show that if an inclusion $1 \in A \subset B$ of unital $C^{*}$ -algebras has index-finite type and finite depth, and $A$ is a simple unital $C^{*}$-algebra with $\mathop{tsr}(A) = 1$ and Property (SP), then $B$ has cancellation. In particular, if $\alpha$ is an action of a finite group $G$ on $A$, then the crossed product $A \rtimes_{\alpha} G$ has cancellation. For outer actions of $\mathbb{Z}$, we obtain cancellation for $A \times_{\alpha} \mathbb{Z}$ under the additional condition that $\alpha_{*} = \text{id}$ on $K_0(A)$. Examples are given.$