Lemma

For every $f\in \mathscr{O}$(Sp$_{A_{(\mathbf{C})}}(x)$), one has $f^*(1\otimes x) =\overline{f(1 \otimes x)}$.

The mappings $f\mapsto f(1\otimes x)$ and $f\mapsto \overline{f^*(1 \otimes x)}$ are continuous unital homomorphisms of $\mathbf{C}$-algebras from $\mathscr{O}$(Sp($x$)) into $A_{(\mathbf{C})}$ which map $z$ to $1\otimes x$; they are therefore equal (I, p. 74, th. 5).