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

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.

• @Ennar. You're right. I thought I had found a different duplicate target. I'm certain I've seen this question here before. – Theo Bendit Mar 6 at 8:20
• Do you mean $ab=ba$ for a particular $a$ and $b$ or for all $a$ and $b$? – MPW Mar 6 at 8:46
• 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. – Arthur Mar 6 at 8:46
• Have a look at Baker-Campbell-Hausdorff formula. – N. Ciccoli Mar 6 at 8:57
• @N. Ciccoli, only one direction is obvious from the formula, right? – Ennar Mar 6 at 9:02