Suppose $\mathscr{M}$ is a properly infinite von Neumann algebra. Can we find a $n\times n$ self-adjoint system of matrix unit whose cardinality is infinite? That is the existence of $\{E_a\}_{a\in I}$ equivalent projections in $\mathscr{M}$ with $\sum_a E_a = I$ where $I$ is infinite.
Actually I wanted to prove that such $\mathscr{M}$ contains a copy of $\mathcal{B}(\mathcal{H})$. For this I thought to find a system of matrix units.
I know that for a von Neumann algebra without type $I$ central summands there is always a copy of $M_n(\mathbb{C})$ in it by Lemma 6.5.6 of Kadison-Ringrose II. But is it possible to get a matrix unit of infinite cardinality in properly infinite case so that I can say that such algebra contains a copy of $\mathcal{B}(\mathcal{H})$.
Any help is appreciated. Thanks.
EDIT Since $I\in \mathscr{M}$ is properly infinite so we can use 'Halving Lemma' (Kadison-Ringrose-II, Lemma 6.3.3) to get two projections $E_1, E_2$ such that $I = E_1 + E_2$ and $E_1 \sim E_2 \sim I$. Now we use Halving lemma on $E_1, E_2$ and their subprojections we can get $\{E_1, E_2, \dots\}$ an infinite family of projections such that $\sum_{i}E_i= I$. Is this right?
There is a stronger version of the halving lemma that gives you directly an infinite sequence of mutually orthogonal projections all equivalent to $I$ and summing up to $I$. To prove it from the weaker form of the halving lemma, though, you need to be more careful. Basically, you can begin by writing $I = E_1 + F_1$ and $E_1 \sim F_1 \sim I$. Then halve $F_1$ to obtain $F_1 = E_2 + F_2$ with $E_2 \sim F_2 \sim I$. The halve $F_2$ and so on. Doing this recursively gives you an infinite sequence of mutually orthogonal projections all equivalent to $I$, but there is no guarantee that they will sum up to $I$. But you can then just change $E_1$ to $E_1 + (I - \sum_i E_i)$. We observe that $E_1 \leq E_1 + (I - \sum_i E_i) \leq I$ but $E_1 \sim I$, so $E_1 + (I - \sum_i E_i) \sim I$, so you get what you want, namely an infinite sequence of mutually orthogonal projections all equivalent to $I$ and summing up to $I$.
After this point you can easily construct a system of matrix unit by defining $v_i$ to be a partial isometry from $E_i$ to $E_1$, for all $i$. These exist because all $E_i$ are equivalent. Then $\{e_{ij}\}_{i, j \in \mathbb{N}}$ where $e_{ij} = v_i^*v_j$ form a system of matrix units, as desired.