Lemma 11

Let $n\geqslant 1$ be an integer. Let $\boldsymbol{a}\in A^n$. For every open neighbourhood U of Sp$^n(\boldsymbol{a})$, there exists an envelope of $(\boldsymbol{a},U)$.

Let $(a_{\lambda})_{\lambda\in\Lambda}$ be a family of elements of A extending the family $\boldsymbol{a}$ and generating the unital Banach algebra A topologically. Let $\pi$ be the canonical projection of $\mathbf{C}^{\Lambda}$ onto $\mathbf{C}^n$ and let $U'=\pi^{-1}(U)$. Then $U'$ is a neighbourhood of Sp$^{\Lambda}((a_{\lambda}))$, and Sp$^{\Lambda}((a_{\lambda}))$ is polynomially convex (I, p. 44, lemma 4). By lemma 6 of I, p. 47, there exists a finite subset $\Lambda_0$ of Λ containing $\{1,2, . . . , n\}$ such that pr$_{\Lambda_0}(U')$ contains the polynomially convex envelope S of pr$_{\Lambda_0}$(Sp$^{\Lambda}((a_{\lambda})_{\lambda\in\Lambda})$) $=$ Sp$^{\Lambda_0}((a_{\lambda})_{\lambda\in\Lambda_0})$. Let $p\geqslant 0$ be the integer such that $\Lambda_0$ has cardinal $n+p$, and let $j$ be a bijection of $\{1, . . . , n+p\}$ onto $\Lambda_0$ which coincides with the identical mapping on $\{1, . . . , n\}$. Since the projection of S is contained in U, the family $(a_{j(k)})_{1\leqslant k\leqslant n+p}$ is an envelope of $(\boldsymbol{a},U)$.