Let $A$ be a Banach algebra we khow that

If $ab=ba$ then $e^{a+b}=e^{a}e^{b}$

my question is

Does $e^{a+b}=e^{a}e^{b}$ implies that $ab=ba$?

Any comment or response is appreciated.

  • $\begingroup$ @Ennar. You're right. I thought I had found a different duplicate target. I'm certain I've seen this question here before. $\endgroup$ – Theo Bendit Mar 6 at 8:20
  • $\begingroup$ Do you mean $ab=ba$ for a particular $a$ and $b$ or for all $a$ and $b$? $\endgroup$ – MPW Mar 6 at 8:46
  • $\begingroup$ Writing out $e^{a+b}-e^ae^b$, I highly doubt that the vanishing of all those terms implies the vanishing of $ab-ba$. But I might be wrong. $\endgroup$ – Arthur Mar 6 at 8:46
  • $\begingroup$ Have a look at Baker-Campbell-Hausdorff formula. $\endgroup$ – N. Ciccoli Mar 6 at 8:57
  • $\begingroup$ @N. Ciccoli, only one direction is obvious from the formula, right? $\endgroup$ – Ennar Mar 6 at 9:02

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