Exercise 4, § 8

Let X be a compact topological space such that the Banach algebra $\mathscr{C}(X)$ is generated by a sequence $(j_n)$ of idempotents. The function

$\infty$ $\omega \mapsto \sum\frac{1}{3^n}(2j_n(\omega )-1)$ $n=1$

separates the points of X. It generates the Banach algebra $\mathscr{C}(X)$.