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 $\pi(a\otimes b)=ab$.