Proposition 33.3.2.label Let $A$ be a unital Banach algebra and $x \in A$. If there exists $\theta \in [0, 2\pi)$ such that
\[\sigma_{A}(x) \subset \complex \setminus e^{i\theta}[0, \infty)\]
- (1)
There exists $y \in A$ such that $x = \exp(y)$.
- (2)
$x \in G_{0}(A)$.
Proof. (1): 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 $x = \exp(y)$.
(2): Proposition 33.6.3.$\square$
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