Left topological zero-divisors in Banach algebras. Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by
$$
\forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|,
$$
where $ \mathbb{S}(A) $ denotes the unit sphere of $ A $.

Definition An element $ a \in A $ is called a left topological zero-divisor iff $ \zeta(a) = 0 $, or equivalently, iff there exists a sequence $ (b_{n})_{n \in \mathbb{N}} $ in $ \mathbb{S}(A) $ such that $ \displaystyle \lim_{n \to \infty} a b_{n} = 0 $.

(a) Prove that a left topological zero-divisor is not invertible.
(b) Prove that $ \zeta: A \longrightarrow [0,\infty) $ is continuous.
(c) Let $ a \in \partial(G(A)) $, the boundary of $ G(A) $. Prove that there exists a sequence $ (\nu_{n})_{n \in \mathbb{N}} $ of invertible elements in $ A $ such that $ \displaystyle \lim_{n \to \infty} \nu_{n} = a $ and $ \displaystyle \lim_{n \to \infty} \| \nu_{n}^{-1} \| = \infty $.
Note: $ G(A) \stackrel{\text{def}}{=} \{ a \in A \mid \text{$ a $ is invertible} \} $.

Hint for part (a): $b_n = a^{-1} (a b_n)$.  
