Example of a unital C* algebra where $conv(proj(\mathcal{A}))$ is not dense

Question

If $\mathcal{A}$ is a unital $C^*$-algebra then what is an an example of a unital $C^*$-algebra $\mathcal{A}$ for which conv Proj$(\mathcal{A})$ is not dense in $\mathcal{P}_1(\mathcal{A}):= \lbrace x \in \mathcal{A^+} : \Vert x \Vert \leq 1 \rbrace$$? Or is it always dense?

Answer 1

If the C*-algebra has few projections, this will be false. Try $A = C([0,1])$. Then the convex hull of projections is just the convex hull of $\{0,1\}$ (since $[0,1]$ is connected), which just gives non-negative constant functions bounded by 1. This clearly isn't all the positive contractions.