What are some examples of Banach algerba $A$ satsifying the following two conditions?
$1$. $A$ does not have an approximate idebtity.
$2$. $A^2=A$. That is, for any $a\in A$, there exist some $b,c\in A$ such that $a=bc$.
A direct application of the Cohen factorization theorem shows that if $A$ has a bounded approximate identity, then $2$ holds.
Any help would be appreciated.