Let $Tr: M_n(\mathbb{C}) \to \mathbb{C}$ be the standard trace given by
$$Tr \begin{pmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & x_{22} & x_{23} & \dots & x_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{n1} & x_{n2} & x_{n3} & \dots & x_{nn} \end{pmatrix} = \sum_{j=1}^n x_{jj}$$
Let $p,q$ be projections in $ M_n(\mathbb{C})$. I have shown that the following are equivalent:
(i) $p \sim q$
(ii) $Tr(p)=Tr(q)$
(iii) $Dim(p(\mathbb{C}^n))=Dim(q(\mathbb{C}^n))$
Now I want to use this to show that $\mathcal{D}(\mathbb{C}) \cong \mathbb{Z}^+$, when $\mathbb{Z}^+$ is equipped with the usual addition.
Proof idea:
As a start I have simply written up the definition of $\mathcal{D}(\mathbb{C})$ from my book such that:
$$\mathcal{D}(\mathbb{C}) \overset{2.3.3}{:=}\big( \mathcal{P}_\infty (\mathbb{C}) \big) / \sim_0 \overset{2.3.1}= \Big( \bigcup_{n=1}^{\infty} \mathcal{P}_n (\mathbb{C}) \Big) / \sim_0 \overset{2.3.1}= \Big( \bigcup_{n=1}^{\infty} \mathcal{P} (M_n (\mathbb{C})) \Big) / \sim_0 $$
Where the unions are disjoint. For each $p$ in $\mathcal{P}_\infty$ we let $[p]_\mathcal{D}$ denote the equivalence class containing $p$.
Now, I want to use the trace as a bijection from $\mathcal{D}(\mathbb{C}) \to \mathbb{Z}^+$ but I am struggling with doing something formal about it. Intuitively I think that if $p$ and $q$ are projections then if their traces are equal then they are equivalent and then they are actually in the same equivalence class so I think this is intuitively why this trace function could be injective. In my book it says that if $p$ and $q$ both belong to $\mathcal{P}_n(A)$ (where $A$ is a $C^*$-algebra) then $p \sim_0 q$ iff $p \sim q$ where $\sim$ is the Murray-von Neuman equivalence, so I think it should be okay like this.
I am having some trouble "seeing" the surjectivity and also why $Tr$ should only take values in $\mathbb{Z}^+$. If $p$ is a projection then the only eigenvalues of $p$ are $0$ and $1$, but how does this help me with the trace? Can I say something about the elements on the diagonal?
Finally, I wondered if this surjectivity problems is because I should do something else or something different.