Exercise 17, § 4

Soit $X_1$ (resp. $X_2$) the set of $(z_1, z_2)\in \mathbf{C}^2$ such that

$\frac{1}{2}\leqslant |z_1|^2+|z_2|^2\leqslant 1$, respectively $|z_1|^2+|z_2|^2\leqslant 1$.

Let $X = X_1\cup \{(0,0)\} \subset \mathbf{C}^2$. Let $A =\mathscr{C}(X)$, and let $a_1, a_2$ be the restrictions to X of the coordinate functions on $\mathbf{C}^2$.

a) The simultaneous spectrum Sp$^2_A(a_1, a_2)$ is equal to X.

b) Construct two distinct continuous unital morphisms from $\mathscr{O}(X)$ into A transforming $z_1$ into $a_1$ and $z_2$ into $a_2$. (Use the following theorem: for every function $f$ holomorphic in a neighbourhood of $X_1$, there exists a function $g$ holomorphic in a neighbourhood of $X_2$ which coincides with $f$ in a neighbourhood of $X_1($cf. I, p. 168, exerc. 14).)