Exercise 18, § 3

Let A be a commutative unital Banach algebra and $x$ an element of A. Suppose that the closed full subalgebra of A generated by $x$ is equal to A, so that the mapping $\chi \mapsto \chi (x)$ allows one to identify $\mathsf{X}(A)$ and Sp$_A(x)$. Let $\mathscr{H}$ be the set of functions $\mathscr{G}y$ on Sp$_A(x)$, for $y$ ranging over A. Then $\check{S}_{\mathscr{H}}$ (Sp$_A(x)$) (INT, IV, §7, n$^o4$) is the boundary of Sp$_A(x)$ relative to $\mathbf{C}$. (To now prove that $\check{S}_{\mathscr{H}}$ (Sp$_A(x)$) is contained in this boundary, use the maximum principle. Conversely, let $z_0$ be a point of this boundary; to now prove that $z_0\in \check{S}_{\mathscr{H}}$ (Sp$_A(x)$), consider $(x-z_1)^{-1}$ for $z_1$ sufficiently close to $z_0$ in $\mathbf{C}-$ Sp$_A(x)$.)