In Banach algebra $A$ find an example such hat $e^{a+b} \not =e^ae^b$

Question

Let $A$ be a Banach algebra

if $ab=ba$ then we have $e^{a+b}=e^ae^b$

without $ab=ba$, I want to find an example such hat $e^{a+b} \not =e^ae^b$

Any help will be greatly appreciated.

Answer 1

Hint: In $\mathbb{R}^{2\times2}$, if $a=\left[\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right]$ and $b=\left[\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\right]$, then what are $e^a$, $e^b$, and $e^{a+b}$?