The following is a part of a theorem in Murphy's C\*-algebras and operator theory:

Let $A$ be a C\*-algebra acting irreducibly on a Hilbert space $H$ and $q$ be a nonzero projection in $A$ of minimal finite rank. Show that $q$ is rank one.

Murphy uses below argument for it:![enter image description here](https://i.stack.imgur.com/hBsd0.png)![enter image description here](https://i.stack.imgur.com/03Xhc.png)

While I think $qAq=\Bbb q$ implies that $q$ is rank one operator, because if $q=x\otimes x + y\otimes y $ for $x,y\in H$, then for $u\in A$, $$(x\otimes x+ y\otimes y) u(x\otimes x+ y\otimes y)= \lambda\_1 x\otimes x + \lambda\_2 x\otimes y + \lambda\_3 y\otimes x + \lambda\_4 y\otimes y \not \in \Bbb Cq $$
Is my argument correct? Where is my mistake?

Also in the earlier part of this book, The writer supposes $\phi:K(H) \to K(H')$ is an $\*-$ isomopphism and $p\_e = e\otimes e \in K(H)$ and put $q\_e=\phi(p\_e)$. He shows that $q\_e K(H) q\_e = \Bbb C q\_e$. Then he claims this is easily inferred from this that $q\_e$ is rank one projection. BUt I do not know how he concludes it. Please give me a hint. Thanks