Example 2

Let $n\geqslant 0$ be an integer. Let $A_n$ be the algebra of functions $f: [0,1]\rightarrow K$ admitting continuous derivatives on $[0,1]$ up to order $n$. Endowed with the norm

$\|f\|=\sum^nk1$! sup$_{0\leqslant t\leqslant 1}|f^{(k)}(t)|$,

$k=0$

it is a Banach algebra (example 4 of I, p. 18). For every $n$, the character space $\mathsf{X}(A_n)$ is identified with $[0,1]$ and $\mathscr{G}$ with the inclusion of $A_n$ in $\mathscr{C}([0,1])$ (cf. example 1 of I, p. 144).