Proposition 4

Let L be a polynomially convex compact subset of $\mathbf{C}^n$ and let U be an open neighbourhood of L. For every function $f\in \mathscr{O}(U; A)$, there exists a sequence of polynomial functions on $\mathbf{C}^n$ with coefficients in A which converges to $f|L$ in $\mathscr{C}(L; A)$.

One may suppose that L is not empty and that A is nonzero. Let P (resp. $P_0$) be the set of restrictions to L of polynomial functions on $\mathbf{C}^n$ with coefficients in A (resp. with coefficients in $\mathbf{C}$). Let B (resp. $B_0$) be the Banach algebra closure of P (resp. of $P_0$) in $\mathscr{C}(L; A)$. Let $\iota$ denote the injection of A onto the normed subalgebra of B consisting of the constant functions.

Let $z_1, . . . , z_n$ be the restrictions to L of the coordinate functions on $\mathbf{C}^n$; they are elements of $B_0$, and, setting $\boldsymbol{z}= (z_1, . . . , z_n)$, one has Sp$^n_{B_0}(\boldsymbol{z}) = L$ by Proposition 15 of I, p. 47.

Let $f\in \mathscr{O}(U; A)$. By composition with $\iota$, the function $f$ defines an element $f_B=\iota \circ f$ of $\mathscr{O}(U; B)$. Since Sp$^n_B(\boldsymbol{z})\subset$ Sp$^n_{B_0}(\boldsymbol{z})\subset U$, one can form the element $b= \Theta_{\boldsymbol{z}}(f_B)$ of B. Let $\boldsymbol{w}= (w_1, . . . , w_n)\in L$, and let $\varphi$ be the continuous unital morphism $g\mapsto g(\boldsymbol{w})$ of B into A. One has $\varphi \circ \iota =$ Id$_A$, so that $\varphi \circ f_B=f$. Since $\varphi (z_i) =w_i$, prop. 3 of I, p. 66 implies $\varphi (\Theta_{\boldsymbol{z}}(f_B)) = \Theta_{\boldsymbol{w}}(\varphi \circ f_B)$. Hence

by the corollary of prop. 2 of I, p. 65. Thus one has $f|L =b$; in particular, $f|L$ belongs to B. This proves the proposition.