Exercise 14, § 3

In $\mathbf{C}^2$, let Ω be the compact set defined by

and let A be the subalgebra of $\mathscr{C}(\Omega )$ formed by the functions which are holomorphic in $\mathring{\Omega}$. Let U be the open set defined by $|z_1|^2+|z_2|^2<1$. By virtue of a theorem of Hartogs (cf. J.-P. Demailly, Complex analytic and differential geometry, Ch. I, th. 3.28) for every $f\in A$, there exists one and only one function $\widetilde{f}\in \mathscr{C}(U)$ which is holomorphic in U and coincides with $f$ on Ω. Now prove that the canonical injection of A into $\mathscr{C}(\Omega )$ is an isometric morphism $h$ whose image is a full subalgebra of $\mathscr{C}(\Omega )$, but $\mathsf{X}(h)$ is not surjective.