0
$\begingroup$

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.

Improve this question
$\endgroup$
0

1 Answer 1

Reset to default
8
$\begingroup$

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 $B^2=B$.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thank you and what about infinite dimensional Banach algebras? Is there any? $\endgroup$
    – Fermat
    Commented Feb 23, 2019 at 13:44
  • 2
    $\begingroup$ 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
    Commented Feb 23, 2019 at 14:03
  • 4
    $\begingroup$ ... or just the direct product of $A$ or $B$ above, with your favorite unital infinite-dimensional Banach algebra. $\endgroup$
    – YCor
    Commented Feb 23, 2019 at 14:58

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .