Exercise 10, § 3

Let $A$ be a commutative Banach algebra and let $D$ be a continuous derivation of $A$ into $A$, that is to say a continuous linear mapping of $A$ into $A$ such that $D(ab) = (Da)b+a(Db)$ for $a, b\in A$.

a) For every $\chi \in \mathsf{X}'(A)$ and every $\lambda \in \mathbf{C}$, the series

$\infty n$

$\varphi_{\lambda}(a) =\sum\lambda n$! $\chi (D^n(a))$

$n=0$

converges$;\lambda \mapsto \varphi_{\lambda}(a)$ is an entire function, and $a\mapsto \varphi_{\lambda}(a)$ is a character of A.

b) The number $\varphi_{\lambda}(a)$ is independent of $\lambda$. (Observe that $|\varphi_{\lambda}(a)|\leqslant \|a\|$ according to a).) Hence $\chi (D(a)) = 0$.

c) The image of D is contained in the radical of A. Deduce a new proof of assertion f ) of Exercise 31 of I, p. 162.