Let $\{M_i\}_{i\in I}$ be a family of von Neumann algebras. Let H denote the direct sum$\Sigma_{i\in I} H_i$ of Hilbert spaces $\{H_i\}_{i\in I}$. Every vector $h=\{h_i\}$ in H is denoted by $\Sigma_{i\in I}h_i$. For each bounded sequence $\{x_i\}$ in $M= \Sigma M_i$, we define an operator x on H by $x\Sigma h_i=\Sigma x_ih_i$.
Why is M a von Neumann algebra?
By the double commutant theorem: $t \in B(H)$ in the commutant $M'$ of $M$ if and only if ($p_i$ denotes the projection onto the $i$-th coordinate)
$$ p_i t p_j = 0, \;\; \forall {i \neq j} $$
and
$$ p_i t p_i \in M_i' \subset B(H_i). $$
Similarly, $s \in M''$ if and only if
$$ p_i s p_j = 0, \;\; \forall {i \neq j} $$
and
$$ p_i s p_i \in M_i'' = M_i\subset B(H_i), $$
i.e. $s \in M$.