Assume That $A$ is a commutative complex Banach algebra. Let $G$ be the connected component of invertible elements containing the identity.



> 
> Is there an smooth embedded curve $c:(-\epsilon, \epsilon) \to A$ with $c(0)=0$ and a neighborhood $W$ of $0$ and a continuous (or Frechet differentiable or holomorphic\* ) map $F:(W \cap G)\setminus c([0, \epsilon)) \to A$ with $\mathrm{exp}\circ F=\mathrm{id}$?
> 
> 
> 


* By holomorphic map we mean a map with local Taylor series around each point of its domain.



Local branch of logarithm in commutative Banach algebras
My main reference here is [Lorch - The Theory of Analytic Functions in Normed Abelian Vector Rings (1942)](https://www.ams.org/tran/1943-054-03/S0002-9947-1943-0009090-0/S0002-9947-1943-0009090-0.pdf) (freely available from AMS).


Basically, in this paper Lorch shows that the classical function theory of a single complex variable can be carried over to the infinite-dimensional setting, i.e. replace $\mathbb{C}$ *everywhere* by a commutative complex Banach algebra $A$. In particular, this includes the notions of a rectifiable curve $\gamma$ taking values in $A$ and of a line integral $\int\_\gamma f(Z)dZ$ for continuous functions $f:A\to A$ (or defined on some subset of $A$ containing the image of $\gamma$). I am using $Z$ instead of $z$ to denote the $A$-valued variable on purpose in order to distinguish from the case of Banach-space-valued holomorphic functions $g:\mathbb{C}\to A$, which are still a different animal.


Lorch notes that $\exp:A\to A$ takes values in $G$ (denoted by $G\_1$ in his paper) and proves that the periods of $\exp$ are given precisely by
$$
2\pi i\sum\_{k=1}^n\pm j\_k,
$$
for $j\_k$ *some* (wlog. non-trivial) idempotents of $A$. In particular, $\exp$ is simply periodic iff $A$ is irreducible.


Finally he defines 
$$
\log(Z):=\int\_1^Z\frac{dX}{X},
$$
where the path from $1$ to $Z$ is an arbitrary one in $U(A)$, the group of units. In particular, $\log$ takes arguments only from $G$. He notes that $\log:G\to A$ is surjective and it's "multi-valued" in accordance with the periods of $\exp$.


