In Banach algebra $A$ if $ab=ba$ prove that $e^{a+b}=e^ae^b$

Question

Let $A$ be a Banach algebra

if $ab=ba$ then prove that $e^{a+b}=e^ae^b$

I've started by $e^a=\sum _{n=0}^{\infty }\frac{a^n}{n!}$, I want to know if this is correct way?

Any help will be greatly appreciated.

Answer 1

Yes. Expand as power series, and you can rearrange the $a$'s and $b$'s because they commute (so $(a+b)^2=a^2+2ab+b^2$ for example), so you can pretend you are doing high-school maths with numbers. You might want to read about the BCH (Baker Campbell Hausdorff) expansion. A slightly harder result holds when $a$ and $b$ both commute with their commutator $[a,b]$, and if that fails then it gets harder ...