Proposition 7

The mapping $f\mapsto f(x)$ is the unique unital morphism of involutive algebras from $\mathscr{C}$ (Sp$_A(x)$) into A such that the identity mapping $z$ of Sp$_A(x)$ has image $x$.

In fact, the unital subalgebra of $\mathscr{C}$ (Sp$_A(x)$) generated by the elements $z$ and $\overline{z}$ of $\mathscr{C}$(Sp$_A(x)$) is dense in $\mathscr{C}$ (Sp$_A(x)$) (TG, X, p. 40, cor. 1). Since every morphism of involutive algebras from $\mathscr{C}$ (Sp$_A(x)$) into A is continuous (I, p. 104, prop. 2), there exists at most one morphism of involutive algebras from $\mathscr{C}$(Sp$_A(x)$) into A which maps $z$ to $x$.

The following corollary shows that when $f$ is the restriction of a holomorphic function in a neighbourhood of Sp$_A(x)$, the definition of $f(x)$ coincides with that of the holomorphic functional calculus in one variable of No. 9 of I, p. 74.