with unit $e_{i_0i_0}$. Then $M$ is generated, as a von Neumann algebra, by $N$ and the matrix units $(e_{ij})_{i,j\in I}$, and there exists a normal $*$-isomorphism
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\begin{align*}
N\overline{\otimes}\mathcal{L}(\ell^2(I))\cong M
\end{align*}
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sending $e_{i_0i_0}xe_{i_0i_0}\otimes E_{ij}$ to $e_{ii_0}xe_{i_0j}$ for all $x\in M$ and all $i,j\in I$. Equivalently, under the inverse isomorphism, each matrix unit $e_{ij}$ corresponds to $1_N\otimes E_{ij}$.