Example 2

Let A be a commutative involutive algebra. Let $a$ be in A. Since Sp$'_A(x)$ is the union of $\{0\}$ and of the image of the Gelfand transform $\mathscr{G}(a)$ (prop. 6 of I, p. 37), the element $a$ is positive if, and only if, the Gelfand transform $\mathscr{G}(a)$ is a positive function.