Exercise 4.7(iv) in the book "An introduction to K-theory for $C^*$-algebras" by M. RØRDAM:
Let A be a $C^*$-algebra and define a relation $\preceq$ on $P_\infty(A)$ as follows. For $p\in P_n(A)$ and $q\in P_m(A)$, $p\preceq q$ if there is a projection $q_0\in P_m(A)$ such that $p\sim_{0}q_0\leq q$ (where $\sim_0$ means the MvN relation on $P_\infty(A)$).
I have shown the following:
(i). $p\preceq q$ iff $q\sim_{0} p\oplus p_0$ for some $p_0\in P_{\infty}(A)$.
(ii). $\preceq$ is transitive.
(iii). If $p_1,p_2,q_1,q_2\in P_{\infty}(A)$ satisfy $p_j\preceq q_j$ for $j=1,2$, then $p_1\oplus p_2 \preceq q_1\oplus q_2$.
I didn't succeed to show:
(iv). Show that a non-zero projection $p\in P_{n}(A)$ is properly infinite iff $p\oplus p \preceq p$.
The definition I use for properly infinite projection is:
A non-zero projection $p$ in a $C^*$-algebra $A$ is said to be properly infinite if there are mutually orthogonal projections $e,f$ in A such that $e\leq p, f\leq p$ and $p\sim e\sim f$.
I have succeed to show the direction $"\Rightarrow"$.
For $"\Leftarrow"$: If $p\oplus p \preceq p$ then let $q\in P_n(A)$ be such that $p\oplus p \sim_{0} q \leq p$. Then $p-q$ and $q$ are mutually orthogonal projections in $P_n(A)$, it could be nice if $p-q \sim p \sim q$ , because then we're done. But I couldn't show that, and I'm not sure it is the case.
Thank you for your time!
I think I have an answer to myself:
For $"\Leftarrow$" : If $p⊕p⪯p$ then let $q∈P_n(A)$ be such that $p⊕p∼_0q≤p$.
Let $v\in M_{2n,n}(A)$ be the rectangle matrix satisfies $vv^*=p\oplus p$ and $v^*v=q$.
Define: $e=v^*(p\oplus 0_n)v$ and $f=v^*(0_n\oplus p)v$, it's not difficult to see that $e,f \in M_n(A)$ are mutually orthogonal projections, and $e\sim f \sim p$.
Thanks to André S., I've found out the importance of $q\leq p$; we use it to show $e\leq p$ and $f\leq p$.