Properties of $ \text{Exp}(A) $, where $ A $ is a Banach algebra. $ \newcommand{\Exp}{\operatorname{Exp}} $ Let $ A $ be a unital Banach algebra. For $ a \in A $, consider
$$
\Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in \mathbb{N} ~ \land ~ a_{1},a_{2},\ldots,a_{n} \in A \}.
$$
I need to show the following:
(a) $ \Exp(A) $ is a subgroup of $ G(A) $, where $ G(A) $ is the set of invertible elements of $ A $.
(b) Prove that for every $ b \in \Exp(A) $, there exists a continuous function $ F: [0,1] \to G(A) $ such that $ F(0) = \mathbf{1}_{A} $ and $ F(1) = b $.
(c) Assume that $ c \in A $ and $ \sigma(c) \subseteq \{ \lambda \in \mathbb{C} ~|~ \text{Re}(\lambda) > 0 \} $. Using complex analysis, show that there exists a ‘logarithm of $ c $ in $ A $’, i.e., $ c = e^{d} $ for some $ d \in A $.
(d) Prove that $ \Exp(A) $ is an open subset of $ A $ using the following guidelines. Let $ b_{0} \in \Exp(A) $. Suppose that $ b \in G(A) $ satisfies $ \| b - b_{0} \| < \| b_{0}^{-1} \|^{-1} $ and consider $ c := b b_{0}^{-1} $. By estimating $ \| c - \mathbf{1}_{A} \| $, prove that Part (c) is applicable to $ c $, so $ b = c b_{0} \in \Exp(A) $.
I have thought about this question for a long time, but I could only figure out Part (a) and have no clue on how to do the rest. Please help. Thank you.
 A: (a) As you have already figured out how to do this part, I shall skip it. :)

(b) Suppose that $ b = e^{a_{1}} \cdots e^{a_{n}} \in \text{Exp}(A) $. Define $ f: [0,1] \to \mathcal{G}(A) $ as follows:
$$
\forall t \in [0,1]: \quad f(t) \stackrel{\text{def}}{=} e^{t a_{1}} \cdots e^{t a_{n}}.
$$
Then


*

*$ f $ is continuous,

*$ f(0) = \mathbf{1}_{A} $ and

*$ f(1) = b $.

(c) As $ \sigma(c) $ is a compact subset of $ \Omega := \{ \lambda \in \mathbb{C} ~|~ \text{Re}(\lambda) > 0 \} $, we can easily find a positively oriented rectifiable Jordan curve $ \gamma: [0,1] \to \Omega $ such that $ \sigma(c) $ lies in the interior region of $ \gamma $. As $ \ln: \Omega \to \mathbb{C} $ is a holomorphic function, we can use the Holomorphic Functional Calculus to define
$$
\ln(c) \stackrel{\text{def}}{=} \frac{1}{2 \pi i} \int_{\gamma} \ln(\zeta) \cdot (\zeta \cdot \mathbf{1}_{A} - c)^{-1} ~ d{\zeta}.
$$
Then $ \ln(c) $ has the property that $ e^{\ln(c)} = c $.

(d) Would you like to try this one on your own? Use the given hints to prove that $ \sigma(c) \subseteq \Omega $.
