Exercise 7, § 5

Let R, $T_1,T_2$ be the subsets formed of $\mathbf{C}^4$ formed by the elements $(z_1, z_2, z_3, z_4)\in \mathbf{C}^4$ such that:

a) The set $X = R\cup T_1\cup T_2$ is polynomially convex. (Now prove that X is the set of the $(z_1, z_2, z_3, z_4)$ such that

$|z_4z^k_2|\leqslant 1$ and $|z^k_1(z_4-z_3^2)|\leqslant 1$ for $k= 1,2, . . .)$.

b) Let A be the closed subalgebra of $\mathscr{C}(X)$ generated by the restrictions to X of the polynomial functions on $\mathbf{C}^4$. Then X identifies with $\mathsf{X}(A)$.

c) Let $\Gamma_1,\Gamma_2\subset \mathbf{C}$ be the circles of center 0 and of radii 1, 2 oriented in the positive sense. Let $\mu_0, \mu_1, \nu$ be the measures on $\Gamma_1,\Gamma_2,\Gamma_1$ defined by the differential forms $z^{-1}dz,-z^{-1}dz, z^{-2}dz$. Let $\mu=\mu_0+\mu_1$, so that $\mu\otimes \nu$ is a measure on $(\Gamma_1\cup \Gamma_2)\times \Gamma_1$. Let B be the set of the $z= (z_1, z_2, z_3, z_4)\in X$ such that $(z_1, z_3)\in (\Gamma_1\cup \Gamma_2)\times \Gamma_1$. The map $\varphi :z\mapsto (z_1, z_3)$ from B into $(\Gamma_1\cup \Gamma_2)\times \Gamma_1$ is a homeomorphism. Let $\nu$ be the measure $\varphi^{-1}(\mu\otimes \nu )$. Now prove that $\nu$ is orthogonal to A.

d) Let $g$ be the function on X which is zero on $R\cup T_1$ and equal to $z_3$ on $T_2$. Then $g \notin A$.

e) The function $g$ coincides with an element of A in a neighbourhood of every $z\in X$.