Approximate identity in $\ell _p$

Question

Show that for $1\leq p<\infty$, $\ell _p$ with multiplication defined by $(a_n)_n(b_n)_n=(a_nb_n)_n$has an unbounded approximate identity but no bounded approximate identity, I don't know how to start the proof

Any help will be greatly appreciated

Answer 1

The identity for $\mathbb{C}$ is $1$, thus the identity for $\ell_p$ with such multiplication should be $(1)_n$ (constant sequence) but this sequence is in $\ell_p$ if and only if $p=\infty$, which is not the case.

On the other hand, such sequence can be approximated by $(x^k)_k\subseteq\ell_p$, where $x^k = (x^k_n)_n$ is given by $$x^k_n=\begin{cases} 1, & \text{ if } n\leq k \\ 0, & \text{ if } n>k, \end{cases} $$ in the sense that $\lim\limits_{k\to\infty}\|x^ka-a\|_p=0$ for $a\in\ell_p$.