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 $x \in A$.
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.