Proposition 5

The data of the mappings $\Theta_{\boldsymbol{a}}$, for $n\geqslant 1$ and $\boldsymbol{a}\in A^n$, constitute a holomorphic functional calculus on A, that is to say that conditions (CF1), (CF2) and (CF3) of I, p. 51 are satisfied.

Let $n\geqslant 1$ be an integer and $\boldsymbol{a}= (a_1, . . . , a_n)\in A^n$. The mapping $\Theta_{\boldsymbol{a}}$ satisfies $\Theta_{\boldsymbol{a}}(z_i) =a_i$ for every $i$ such that $1\leqslant i\leqslant n$ by lemma 10 of I, p. 64, which proves property (CF2). Lemma 9 of I, p. 63 implies property (CF3) of the mappings $\Theta_{\boldsymbol{a}}$.

The mapping $\Theta_{\boldsymbol{a}}$ is A-linear and continuous (I, p. 61, n$^o5$). It satisfies $\Theta_{\boldsymbol{a}}(1) = 1$ (lemma 9 of I, p. 63). To verify condition (CF1), it remains to establish that $\Theta_{\boldsymbol{a}}$ is an algebra morphism. For this, we shall prove that $\Theta^U_{\boldsymbol{a}}$ is an algebra morphism for every open neighbourhood U of Sp$^n(\boldsymbol{a})$.

Suppose first that U contains the polynomially convex hull K of Sp$^n(\boldsymbol{a})$. Let $f_1$ and $f_2$ be elements of $\mathscr{O}(U; A)$. There exists a sequence $(f_{1,k})$ (resp. $(f_{2,k})$) of polynomial functions converging to $f_1$ (resp. to $f_2$) in $\mathscr{O}(K; A)$ (theorem 2 of I, p. 68), hence in $\mathscr{O}$(Sp$^n(\boldsymbol{a}); A$). For every integer $k$, one has

by lemma 10 of I, p. 64, whence $\Theta^U_{\boldsymbol{a}}(f_1)\Theta^U_{\boldsymbol{a}}(f_2) = \Theta^U_{\boldsymbol{a}}(f_1f_2)$ on passing to the limit.

Consider the general case. Let $\boldsymbol{a}'\in \mathbf{C}^{n+p}$ be an envelopment of $(\boldsymbol{a},U)$ (lemma 11) and $\pi : U\times \mathbf{C}^p\rightarrow U$ the canonical projection. Since $U\times \mathbf{C}^p$ contains the polynomially convex hull of Sp$^{n+p}(\boldsymbol{a}')$, one has

for $f_1$ and $f_2$ in $\mathscr{O}(U; A)$ by the first case. Since, for every function $f\in \mathscr{O}(U; A)$, one has $\Theta^{U\times\mathbf{C}^p}_{\boldsymbol{a}'}(f\circ \pi ) = \Theta^U_{\boldsymbol{a}}(f)$ (condition (CF3) proved above), the conclusion follows, and hence condition (CF1).

We can now prove theorem 1 of I, p. 51. Prop. 5 shows that the family of mappings $(\Theta_{\boldsymbol{a}})_{\boldsymbol{a}}$ is a holomorphic functional calculus on A. It therefore only remains to establish the uniqueness of the holomorphic functional calculus on A.

Let $(\Psi_{\boldsymbol{a}})_{\boldsymbol{a}}$ be a family of mappings defined for every integer $n\geqslant 1$ and every $\boldsymbol{a}\in A^n$ and satisfying conditions (CF1), (CF2), (CF3) of the holomorphic functional calculus on A (I, p. 51). It suffices to prove that for every integer $n\geqslant 1$, for every $\boldsymbol{a}\in A^n$ and for every open neighbourhood U of $\boldsymbol{a}$, one has $\Theta^U_{\boldsymbol{a}}= \Psi^U_{\boldsymbol{a}}$.

Let $n\geqslant 1$ and $\boldsymbol{a}= (a_1, . . . , a_n)\in A^n$. Let U be an open neighbourhood of Sp$^n(\boldsymbol{a})$. Suppose first that U contains the polynomially convex hull K of Sp$^n(\boldsymbol{a})$. The morphisms $\Theta^U_{\boldsymbol{a}}$ and $\Psi^U_{\boldsymbol{a}}$ coincide on polynomial functions by properties (CF1) and (CF2). By the corollary to theorem 2 of I, p. 68 and the continuity property (CF1), these morphisms are therefore equal.

Let us prove the general case. Let $\boldsymbol{a}'\in \mathbf{C}^{n+p}$ be an envelopment of $(\boldsymbol{a},U)$ and $\pi : U\times \mathbf{C}^p\rightarrow U$ the canonical projection. One has

by property (CF3) and the preceding case. This concludes the proof of theorem 1 of I, p. 51.

Let us remark that theorem 2 of I, p. 68 also entails the following uniqueness result: