Proposition 6

Let $\boldsymbol{a}\in A^n$. Suppose that Sp$^n(\boldsymbol{a})$ is polynomially convex. Let $z_1, . . . , z_n$ be the germs in a neighbourhood of Sp$^n(\boldsymbol{a})$ of the coordinate functions on $\mathbf{C}^n$. Then the mapping $\Theta_{\boldsymbol{a}}$ is the unique continuous unital algebra morphism $\varphi$ from $\mathscr{O}$(Sp$^n(\boldsymbol{a}); A$) into A such that $\varphi (z_1) =a_1, . . . , \varphi (z_n) =a_n$.

Lemma 10 of I, p. 64 and the corollary to proposition 2 of I, p. 65 justify the following notations for the holomorphic functional calculus. Let $n\geqslant 1$ and $\boldsymbol{a}\in A^n$. For every germ $f\in \mathscr{O}(K; A)$ (resp. for every holomorphic function $f\in \mathscr{O}(U; A)$ on an open neighbourhood U of Sp$^n(\boldsymbol{a})$), one sets

This notation is consistent with the notation introduced in A, IV, p. 4, n$^o3$, if $f$ is a polynomial, by properties (CF1) and (CF2).

Properties (CF2) and (CF3) of I, p. 51 may then be written