Why is $GL(B)$ a Banach Lie Group?

Question

Banach Lie Groups are what you'd expect:

https://www.encyclopediaofmath.org/index.php/Lie_group,_Banach

If $B$ is a Banach algebra then why is $GL(B)$, the set of invertible elements of $B$, a Banach Lie group?

Answer 1

This follows from three simple observations:

1. The subset $\operatorname{GL}(B) \subset B$ is open, so it is a Banach manifold modeled on $B$ itself.

2. Multiplication is the restriction of a continuous linear map, hence it is analytic.

3. Inversion is locally given by the Neumann series, hence it is analytic, too.