Example 3

For $x$ to be unitary, it is necessary and sufficient that its spectrum be contained in the unit circle of $\mathbf{C}$. In fact, let $f\in \mathscr{C}$ (Sp$_A(x)$) be the function defined by $f(z) =zz-1$; the element $x$ is unitary if and only if $f(x) = 0$, that is, if $f$ is zero.