Simple question concerning direct sums and the weak-operator topology Let $U$ be a C$^{*}$ algebra and $\pi_{a}:U\rightarrow B(\mathcal{H_{a}})$ be a representation for each $a\in I$. I want to show that $\bigoplus_{a}\overline{\pi_{a}(U)}$ is the ultraweak closure of the set $\{\oplus_{a} \pi_{a}(A_{a}):\;$sup$\;||A_{a}||<\infty\}$.
If $B\in\bigoplus_{a}\overline{\pi_{a}(U)}$, then $B=\oplus_{a}B_{a}$ where sup$\;||B_{a}||=:r<\infty$. Since $B_{a}\in(\overline{\pi_{a}(U)})_{r}=\overline{(\pi_{a}(U))_{r}}=\overline{(\pi_{a}(U_{r}))}$ for each $a\in I$, there is a net $\{A^{(\gamma)}_{a}\}_{\gamma\in J(a)}\subset U_{r}$ such that $\{\pi_{a}(A^{(\gamma)}_{a})\}$ weak-operator converges to $B_{a}$. Now, I'd like to take the direct sum of all the nets $\{\pi_{a}(A^{(\gamma)}_{a})\}$ and show that this weak-operator converges to $B$. The problem is each of these nets has a different index set $J(a)$ depending on $a$ and I have never come across a situation where I need to take some kind of "direct product" of nets. Is this indeed the right approach?
 A: I think this approach works. Let $J=\prod_{a}J(a)$. Then $J$ is a directed poset with respect to the relation given by $i\leq j\;$ if $\;i(a)\leq j(a)$ for all $a\in I$. We want to show that $\{\oplus_{a}\pi_{a}(A_{a}^{j(a)})\}_{j\in J}$ weak operator converges to $B$. Let $\epsilon>0$ and $\oplus_{a}x_{a}$,$\;\oplus_{a}y_{a}\in\bigoplus_{a}\mathcal{H}_{a}$. Let $\{a_{n}\}_{n\in\mathbb{N}}\subset I$ be a sequence of those indices $a\in I$ such that both $x_{a}\neq 0$ and $y_{a}\neq 0$. Then for each $n\in \mathbb{N}$ $\;\exists\;\beta(n)\in J(a_{n})$ such that $|\langle(\pi_{a_{n}}(A_{a_{n}}^{\gamma(a_{n})})-B_{a_{n}})x_{a_{n}},y_{a_{n}}\rangle|<\epsilon/2^n$ whenever $\gamma(a_{n})\geq \beta(n)$. Hence if $j_{0}\in J$ such that $j_{0}(a_{n})=\beta(n)$ for all $n$, then $|\langle(\oplus_{a}\pi_{n}(A_{a}^{j(a)})-B)\oplus_{a}x_{a},\oplus_{a}y_{a}\rangle|\leq\sum_{n}|\langle(\pi_{a_{n}}(A_{a_{n}}^{j(a_{n})})-B_{a_{n}})x_{a_{n}},y_{a_{n}}\rangle|\leq\epsilon$ whenever $j\geq j_{0}$.
Does this look good?
