Example

Let K $\subset \mathbf{C}^{\Lambda}$ be a compact subset. Let $z= (z_{\lambda})_{\lambda\in\Lambda}$ be the family in $\mathscr{C}(K)$ of the restrictions to K of the coordinate functions of $\mathbf{C}^{\Lambda}$. Then the joint spectrum Sp$^{\Lambda}_{\mathscr{C}(K)}(z)$ is equal to K. In fact, by Corollary 2 of I, p. 32, every character $\chi$ of $\mathscr{C}(K)$ is of the form $f\mapsto f(x)$ for an element $x\in K$, and one then has $(\chi (z_{\lambda}))_{\lambda\in\Lambda}=x$.