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. 



Banach algebra $A$ without an approximate identity but $A^2=A$
For finite-dimensional algebra, to have an approximate unit is the same as having a unit.


The non-unital algebra $A$ of matrices $\begin{pmatrix}0 & x\\ 0 & y\end{pmatrix}$ has no unit (although it has a right unit), so over the reals has no approximate unit, and satisfies $A^2=A$.


The non-unital algebra $B$ of matrices $\begin{pmatrix}a & x & z\\ 0 & 0 & y\\ 0 & 0 & b\end{pmatrix}$ has no unit (and has no left or right unit), so over the reals has no approximate unit, and satisfies $B^2=B$.


