Let $A$ be a $C^\*$-algebra, we denote with $V(A)$ the semigroup of Murray-von Neumann equivalence classes of projections in matrices over $A$ (as usual). In <https://arxiv.org/pdf/math/0310340.pdf>, above definition 5.1. in section 5 (on page 20), there is mentioned the following fact: 


If $A$ is a $C^\*$-algebra and $p$ a projection in $A$, and $n\in\mathbb{N}\_{\ge 1}$, then $nx=[p]$ has a solution $x\in V(A)$ if and only if there is a unital embedding $M\_n\to pAp$. $(\star)$


I need help to prove the direction $\Rightarrow$. 
(The reason why I take a closer look to this statement is that I need $\star$ for an other proof.)


To prove that there is such a unital embedding, one strategy could be (if I'm not mistaken): Find mutually orthogonal projections $f\_1,\cdots f\_n$ in pAp, with $f\_1\sim f\_2 \sim \cdots \sim f\_n$ (where $\sim$ is Murray-von Neumann equivalence) and such that $1=f\_1+ \cdots +f\_n$. Then you can extend the set $\{f\_1,\cdots,f\_n\}$ to matrix units of $M\_n$.


Now, if $nx=[p]$ for an $x\in V(A)$. Write $x=[x']$ for a projection $x'\in M\_m(A)$ for some natural number $m$. Then $\operatorname{diag}(x',x',\cdots,x')\sim\_0 p.$ I.e. if $p$ is a projection in $M\_k(A)$, then there exists a partial isometry $v\in M\_{k,nm}$ such that $\operatorname{diag}(x',x',\cdots,x')=vv^\*$ and $p=v^\*v$ (it's Murray-von Neumann equivalence in different sized matrices). But I don't know how to get these projections $f\_i$ (from here)...


How to get these projections $f\_i$/ or can you provide an alternative approach to prove it/ or can you provide a reference for a proof of $\star$? Thank you.



unital embedding into the coner $C^*$-algebra
Using your notation, we have diag$(x',...,x')\in M\_n\otimes M\_m(A).$ Then $M\_n$ is isomorphic to the algebra (let's call it $B$) generated by $e\_{ij}\otimes x'$ where $e\_{ij}$ are matrix units for $M\_n.$ Since $e\_{ij}\otimes x'$ (and hence everything in $B$) commutes with $vv^\*$, the map from $B$ to $pAp$ defined by $b\mapsto v^\*bv$ is a \*-homomorphism that takes the unit of $B$ (which is diag$(x',...,x')$) to $p.$


