I am trying to show the following which is stated in Exercise 10.11.10 of Blackadars book on K-theory for operator algebras.
A unital, simple, nuclear, stably finite, infinite dimensional C*-algebra with $K_0(A)=\mathbb{Z}$ and $K_1(A)=0$ has a minimal projection.
I don't know how to approach this if anyone could give me some hints.
By proposition 6.3.3 of Blackadar $(K_0(A),K_0(A)_+)$ is an ordered group. The scale $\Sigma(A)$ is the image of Proj(A) in $K_0(A)$. Since $A$ is unital, the scale is simply the elements of $K_0(A)_+$ which are $\le [1_A]$, so the minimal element of the scale is the image of a minimal projection. The scale cannot be zero, since that would imply that $A$ is contractil, which implies $K_0(A)=0$.