Exercise 10, § 4

Let Δ be the disk $|z|\leqslant 1$ in $\mathbf{C}$. Let A be the space of functions $f$ which are continuous in Δ and holomorphic in the interior of Δ, and such that there exists a summable sequence $(c_n)$ of complex numbers satisfying

for every $z\in \Delta$. We then set $\|f\|=\sum_n|c_n|$.

a) Endowed with the multiplication $(f, g)\mapsto f g$, the space A is a unital Banach algebra and $z$ is a topological generator of A.

b) The space $\mathsf{X}(A)$ is identified with Δ; if $f\in A$ and if $g$ is holomorphic in an open neighbourhood of $f(\Delta )$, then $g\circ f\in A$.