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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.

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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 $B^2=B$.

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  • $\begingroup$ Thank you and what about infinite dimensional Banach algebras? Is there any? $\endgroup$ – Fermat Feb 23 '19 at 13:44
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    $\begingroup$ Fix a Euclidean norm on $A$ for which $\|ab\|\le\|a\|\|b\|$ for all $a\in A$ (restricted direct product), consider $\bigoplus_{n\in\mathbf{N}}A$, and complete to a Hilbertian norm. $\endgroup$ – YCor Feb 23 '19 at 14:03
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    $\begingroup$ ... or just the direct product of $A$ or $B$ above, with your favorite unital infinite-dimensional Banach algebra. $\endgroup$ – YCor Feb 23 '19 at 14:58

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