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 $ exp\circ F=Id)$ ?
- By holomorphic map we mean a map with local Taylor series around each point of its domain.
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