Proposition 15

Let Λ be a set. Let K be a polynomially convex compact subset of $\mathbf{C}^{\Lambda}$. Let $A_0$ be the set of the restrictions to K of polynomial functions on $\mathbf{C}^{\Lambda}$, and let A be the closure of $A_0$ in the algebra $\mathscr{C}(K)$.

Let ev$: K\rightarrow \mathsf{X}(A)$ be the mapping defined by $x\mapsto$ ev$_x$, where ev$_x$ is the character $f\mapsto f(x)$ of A. Let $z= (z_{\lambda})_{\lambda\in\Lambda}$ be the family in A of the restrictions to K of the coordinate functions on $\mathbf{C}^{\Lambda}$, and let $\varphi$ be the surjective mapping of $\mathsf{X}(A)$ onto Sp$^{\Lambda}_A(z)$ defined by the family $z$.

Then we have K = Sp$^{\Lambda}_A(z)$, and the mappings ev : K $\rightarrow \mathsf{X}(A)$ and $\varphi :\mathsf{X}(A)\rightarrow K$ are reciprocal homeomorphisms.

The mapping $\varphi \circ$ ev is the identity mapping of K. In particular, K is contained in the image Sp$^{\Lambda}_A(z)$ of $\varphi$.

Proposition 14 implies that Sp$^{\Lambda}_A(z)$ is the polynomially convex hull of Sp$^{\Lambda}_{\mathscr{C}(K)}(z)$. Since, on the other hand, we have Sp$^{\Lambda}_{\mathscr{C}(K)}(z) = K$ (I, p. 42, example), which is polynomially convex by assumption, we deduce that Sp$^{\Lambda}_A(z) = K$.

Since the unital subalgebra generated by the family of elements $z_{\lambda}$ is dense in A, the mapping $\varphi$ is a homeomorphism of $\mathsf{X}(A)$ onto Sp$^{\Lambda}_A(z) = K$ (Proposition 12 of I, p. 43, a)). The identity $\varphi \circ$ ev = Id$_K$ then shows that ev is the reciprocal homeomorphism of $\varphi$.

For every subset $\Lambda '$ of Λ, we denote by pr$_{\Lambda'}$ the canonical projection $\mathbf{C}^{\Lambda}\rightarrow \mathbf{C}^{\Lambda'}$. Let W be a subset of $\mathbf{C}^{\Lambda}$ and V its polynomially convex hull. Put $W'=$ pr$_{\Lambda'}W$. Since every element of $\mathbf{C}[(X_{\lambda})_{\lambda\in\Lambda'}]$ is identified with an element of $\mathbf{C}[(X_{\lambda})_{\lambda\in\Lambda}]$, the polynomially convex hull of $W'$ is contained in pr$_{\Lambda'}V$.