Proposition 29.3.2.label Let $A$ be a unital Banach algebra and $x \in A$. If there exists $\theta \in [0, 2\pi)$ such that
\[\sigma_{A}(x) \subset \complex \setminus e^{i\theta}[0, \infty)\]
then $x \in G_{0}(A)$.
Proof. Since there exists a branch of the logarithm $\ell: \complex \setminus e^{i\theta}[0, \infty) \to \complex$, there exists $y \in A$ such that $x = \exp(y)$. In which case, $x \in G_{0}(A)$ by Proposition 29.5.3.$\square$
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