I have been told, that there is a theorem about irreducible representation of $C^*$-algebras, but I have troubles finding it. It is also possible, that this theorem is consequence of some theorem I've already come across, but I did not realize that. From what I have understand, the theorem should state something like this:
Let $\pi: A \to B(\mathscr{H})$ be a representation. If this representation is irreducible, than for every two points in $\mathscr{H}$ there is a path from one to the other.
If needed I can provide the definitions of irreducible representations or some equivalent theorems.
The Kadison Transivity theorem is just your answer. See for example (Murphy page 150).