Let $A \subset B(H)$ be a non-degenerate $C^*$-algebra and assume $h$ is a non-zero positive operator in $Z(A)$, the center of $A$. Is is then true that the support projection of $h$ is the identity in $B(H)$ ?
It would suffice to prove that $h$ has dense range or that $h$ is injective.
Consider $\mathbb C\oplus\mathbb C\subset M_2(\mathbb C)$, and take $h=(1,0)$. Then $h$ is a projection, and so it is its own support.