Exercise 20, § 3

Let Δ be the disc $|z|\leqslant 1$ in $\mathbf{C}$, A the set of the $f\in \mathscr{C}(\Delta )$ which are holomorphic in the interior of Δ and $A_1$ the Banach subalgebra of $\mathscr{C}(\Delta )$ generated by A and by the function $z\mapsto |z|$. Now prove that $\mathsf{X}(A_1)$ is homeomorphic to the set of the $(x_1, x_2, x_3)\in \mathbf{R}^3$ such that $x^2_1+x^2_2\leqslant x^2_3$ and $0\leqslant x_3\leqslant 1$.