Proposition 12

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

The elements $z$ and $\overline{z}$ of $\mathscr{C}'$(Sp$'_A(x)$) generate a dense subalgebra of $\mathscr{C}'$(Sp$'_A(x)$) (cf. TG, X, p. 40, cor. 2). Since every morphism of involutive algebras from the stellar algebra $\mathscr{C}'$(Sp$'_A(x)$) into the stellar algebra A is continuous (I, p. 104, prop. 2), the result follows.

The results of the preceding number concerning the functional calculus extend to the general case. We shall merely state them and leave it to the readers to complete the proofs, mutatis mutandis.