Let $I$ isbe an arbitary index set, $((A_i)_i,\|.\|_i)$$((A_i)_i,\|.\|_i)_{i\in I}$ be a family of Banach algebras, with approximate diagonal $(m^{i}_α)_α\subseteq A_i\hat\otimes A_i$, and $B=\{(x_i)_i\in {\displaystyle \prod _{i}}A_i|\sum_i\|x_i\|_i<\infty\}$, where $I$ is an arbitary index set$B=\{(x_i)_i\in {\displaystyle \prod _{i\in I}}A_i|\, \sum_i\|x_i\|_i<\infty\}$.
Is there any approximate diagonal for $B$?
My idea:
For finite subset
$F\subseteq I$ we define
$E_F$ as below
$$(E_F)_{i}=\Big\{^{m_\alpha^{i}~~{~i\in F}}_{0~~\mbox{ else }}$$$$
(E_F)_{i}=\Big\{^{m_\alpha^{i}~~{~i\in F}}_{0~~\mbox{ else. }}
$$
thenThen $(E_F)_{F}$ is an approximate diagonal for $B$.,
where partial order is definddefined as $F_1\preceq F_2$ if and only if $ F_1\subseteq F_2$.
Am isI right? ifIf 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 $\pi\colon B\hat\otimes B\to B$ defindedis defined by $\pi(a\otimes b)=ab$.
