# Banach algebra $A$ without an approximate identity but $A^2=A$

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.

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 $$B^2=B$$.

• Thank you and what about infinite dimensional Banach algebras? Is there any? – Fermat Feb 23 '19 at 13:44
• 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. – YCor Feb 23 '19 at 14:03
• ... or just the direct product of $A$ or $B$ above, with your favorite unital infinite-dimensional Banach algebra. – YCor Feb 23 '19 at 14:58