Let $I$ be an arbitary index set, $((A_i)_i,\|.\|_i)_{i\in I}$ be a family of Banach algebras, with approximate diagonal $(m^{i}_α)_α\subseteq A_i\hat\otimes A_i$, and $B=\{(x_i)_i\in {\displaystyle \prod _{i\in I}}A_i|\, \sum_i\|x_i\|_i<\infty\}$. Is there any approximate diagonal for $B$?

**My idea:**
For finite subset
$F\subseteq I$ we define
$E_F$ as below

$$
(E_F)_{i}=\Big\{^{m_\alpha^{i}~~{~i\in F}}_{0~~\mbox{ else. }}
$$
Then $(E_F)_{F}$ is an approximate diagonal for $B$,
where partial order is defined as $F_1\preceq F_2$ if and only if $ F_1\subseteq F_2$.
Am I right? If yes, how can I prove that $\pi((E_F)_{i})a-a\to 0$ in norm?

Here, $\hat{\otimes}$ denotes the projective tensor product of Banach spaces and $\pi\colon B\hat\otimes B\to B$ is defined by $\pi(a\otimes b)=ab$.