Factorization of rank one operators in $A \otimes B(H)$ The following assertion has been utilized several times in a text! Could anyone please tell me a solution or  reference?
Let $A$ be a unital C*-algebra and let $H$ be a Hilbert space. If $a \in A$, $u \in H$ and
$b \in A \otimes B(H)$ satisfy $0 \leq b \leq a \otimes |u\rangle \langle u|$, then
$b = x \otimes |u\rangle \langle u|$ for some $x \in A$.
 A: Let $p=|u\rangle\langle u|$. If we assume that $u$ is a unit vector, then $p$ is a rank-one projection.
We have
$$
0\leq \big(1\otimes (1-p)\big)\,b\,\big(1\otimes (1-p)\big)
\leq \big(1\otimes (1-p)\big)\,(a\otimes p)\,\big(1\otimes (1-p)\big)=0.
$$
Thus $\big(1\otimes (1-p)\big)\,b\,\big(1\otimes (1-p)\big)=0$; as $b\geq0$, this implies
$
b\,\big(1\otimes (1-p)\big)=0
$. So
$$\tag1
b=b(1\otimes p).
$$
If
$$\tag2
b=\lim_n\sum_ja_{jn}\otimes h_{jn}, 
$$
by $(1)$ we can write
$$
b=\lim_n\sum_ja_{jn}\otimes h_{jn}
=\lim_n\sum_ja_{jn}\otimes ph_{jn}p
=\lim_n\sum_j\lambda_{jn}a_{jn}\otimes p
=\lim_n\Big(\sum_j\lambda_{jn}a_{jn}\Big)\otimes p.
$$
Since
$$
\Big\|\Big(\sum_j\lambda_{jn}a_{jn}\Big)\otimes p\Big\|
=\Big\|\Big(\sum_j\lambda_{jn}a_{jn}\Big)\Big\|\,\|p\|
=\Big\|\sum_j\lambda_{jn}a_{jn}\Big\|,
$$
we get from the existence of the limit in $(2)$ that the sequence of sums is Cauchy and thus convergent in $A$. So there exists $x\in A$ with $$b=x\otimes p.$$
When $|u\rangle$ is not unital, we can scale it and repeat the argument with $b$ replaced by a scalar multiple.
