Please help me with the following question.

What are some examples of Banach algebra $A$ satisfying the following two conditions?

$1$.$ A $ does not have an approximate identity.

$2$. $A^2=A$. That is, for any $a∈A$, there exist some $b,c∈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.

Thank you.