Cancellation for inclusions of C*-algebras of finite depth

Abstract

Let 1 ∈ A ⊂ B be a pair of C* - algebras with common unit. We prove that if E: B → A is a conditional expectation with index-finite type and a quasi-basis of n elements, then the topological stable rank satisfies tsr(B) ≤ tsr(A) + n - 1. As an application we show that if an inclusion 1 ∈ A ⊂ B of unital C*-algebras has index-finite type and finite depth, and A is a simple unital C * -algebra with tsr(A) = 1 and Property (SP), then B has cancellation. In particular, if α is an action of a finite group G on A, then the crossed product A ×α G has cancellation. For outer actions of Z, we obtain cancellation for A ×α Z under the additional condition that α * = id on K 0 (A). Examples are given.