Systems of matrix units inside a von Neumann algebra Suppose $\{e_{ij}: i,j=1,2,3....\}$ is a system of matrix units in a von Neumann algebra $M$ how could I show that $M$ is unitarily equivalent to $e_{11}Me_{11}\otimes B(H)$ where $H$ is a seperable hilbert space. How would I even construct this unitary?
This is stated in Vaughan Jones notes on von Neumann algebras but I am struggling to show this.
 A: To take the "$k,j$" element to the 1,1 coordinate you need to do $e_{1k}xe_{j1}$. So the unitary should map
$$
x\longmapsto e_{1k}xe_{j1}\otimes E_{kj}.
$$
To actually construct the unitary, you need to think that $M\subset B(K)$, and you write $K=\bigoplus_j e_{jj}K$. This allows you to construct a unitary $V:K\to e_{11}K\otimes H$ by
$$
V(\sum_j e_{jj}\xi_j)=\sum_j e_{1j}\xi\otimes f_j,
$$
where $\{f_j\}_j$ is an orthonormal basis for $H$. It is easy to check that $V$ above is isometric (so well defined and with closed range, and that its range contains an orthonormal basis of $e_{11}K\otimes H$.
Now, for any $x\in M$,
\begin{align}
Vx\sum_{\ell}e_{\ell\ell}\xi_j
&=V\sum_{\ell}xe_{\ell\ell}\xi_j
=V\sum_{\ell}\sum_{k,j} e_{kk}xe_{jj}e_{\ell\ell}\xi_j
\\
&=V\sum_{k,j} e_{kk}xe_{jj}\xi_j\\
&=\sum_k e_{1k}\sum_jxe_{jj}\xi_j\otimes f_k\\
&=\sum_{k,j}e_{1k}xe_{j1}\,e_{1j}\xi_j\otimes f_k\\
&=\sum_{k,j}(e_{1k}xe_{j1}\otimes E_{kj})\sum_\ell e_{1\ell}\xi_\ell\otimes f_\ell\\
&=\sum_{k,j}(e_{1k}xe_{j1}\otimes E_{kj})V\sum_\ell e_{\ell\ell}\xi_\ell.
\end{align}
Thus
$$
V^*xV=\sum_{k,j}e_{1k}xe_{j1}\otimes E_{kj}
$$
