We discuss three basic properties for C*-algebras, Property (SP), topological stable rank, and cancellation property.

Let 1 ∈ A ⊂ B be an inclusion of C*-algebras with common unit and

E : B → A be a faithful conditional expectation with index finite

type. Assume that A is simple and has Property (SP). Then we have

(a) B has Property (SP).

(b) tsr(B) ≤ 2 if B ⊂ A is of depth 2 and tsr(A) = 1.

As an application

if B ⊂ A is of finite depth and A has cancellation property with

tsr(A) = 1, then B has cancellation