Example of a Banach algebra with given property

Question

Please let me know what are some examples of Banach algerba $A$ for which $A^2=A$ ( that is, $a^2=a$ for each $a\in A$). A direct application of the cohen factorization theorem shows that if $A$ has an approximate identity, then $A$ satisfies this condition. But what are examples of Banach algerbas with this property but not without an approximate identity.

Thank you.

Answer 1

There is no such Banach algebra except $\{0\}$. If $a =a^{2}$ then $\|a\| =\|a^{2}\|\leq \|a\|^{2}$ so $\|a\| \geq 1$ whenever $a \neq 0$ which is absurd.

Answer 2

If $a^2=a$ for all $a \in A$, then $t^2a=t^2a^2=(ta)^2=ta$ for all $a \in A$ and all $t \in \mathbb R.$ Hence

$ta=a$ for all $a \in A$ and all $t \ne 0.$ With $t \to 0$ we get $a=0.$