= Solution
The exponential series is absolutely norm-convergent, since $\sum_{n\ge1}\|x^n\|/n!\le e^{\|x\|}-1$. <Completeness> defines the <Banach algebra exponential> $e^x\in A$. If $xy=yx$, absolute convergence permits rearrangement of the product series, and the binomial identity gives
$$
e^xe^y=\sum_{n=0}^\infty\sum_{j=0}^n\frac{x^jy^{n-j}}{j!(n-j)!}=\sum_{n=0}^\infty\frac{(x+y)^n}{n!}=e^{x+y}.
$$
Taking $y=-x$ yields \b[$(e^x)^{-1}=e^{-x}$].
For the spectral hypothesis, put $K=\sigma_A(u)$. Its complement has an <open>, hence <path-connected>, unbounded component containing $0$. Join $0$ to an exterior point by a simple polygonal arc in that component, and continue it to infinity, avoiding $K$. Removing this polygonal slit gives a <simply connected> <open> neighborhood $\Omega$ of $K$ on which $z$ has a <branch of the complex logarithm> $L(z)$. The <holomorphic functional calculus> therefore defines $x=L(u)$ and its composition rule gives
$$
\boxed{e^x=(e^L)(u)=u}.
$$
The relevant slit is chosen through the resolvent, not assumed to be a fixed negative-real-axis cut. This is the <logarithm from a spectral slit in a Banach algebra>.
Let $H$ be the <subgroup> of finite products of exponentials. It is a <subgroup> because $e^ae^{-a}=1$ and reversing a product gives its inverse. Each product is joined to $1$ by $t\mapsto e^{ta_1}\cdots e^{ta_m}$, so $H\subseteq G_0$. Conversely, if $\|v-1\|<1$, the convergent logarithm series
$$
\log v=\sum_{n=1}^\infty\frac{(-1)^{n+1}(v-1)^n}{n}
$$
satisfies $e^{\log v}=v$, by the scalar analytic identity and functional calculus. Thus $H$ contains a neighborhood of $1$. Its translates make $H$ <open>, and all other cosets are <open> too, making $H$ <closed>. <Connectedness> forces $G_0\subseteq H$. Finally $ge^ag^{-1}=e^{gag^{-1}}$, so $H$ is normal. Therefore
$$
\boxed{G_0=H=\{e^{a_1}\cdots e^{a_m}:m\ge0,\ a_j\in A\}},
$$
an open-and-closed <normal subgroup>. This is the <identity component of Banach-algebra invertibles>; finite products are essential and are not asserted to be single exponentials.
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