Proposition 4

Let X be a compact space. Let B be a unital Banach subalgebra of $\mathscr{C}(X)$, endowed with the induced norm and separating the points of X. We identify X with a closed subset of $\mathsf{X}(B)$ (Prop. 1 of I, p. 142, d)). Let $(x_{\lambda})_{\lambda\in\Lambda}$ be a family of elements of B generating topologically the unital algebra B. Consider the commutative diagram

X $\rightarrow \mathsf{X}(B)$

Sp$_{\mathscr{C}(X)}((x_{\lambda}))\rightarrow$ Sp$_B((x_{\lambda}))$

where $\varphi$ and $\varphi '$ are the mappings defined by the family $(x_{\lambda})_{\lambda\in\Lambda}($cf. No.$^o6$ of I, p. 41), and $i$ and $j$ are the canonical injections. Then:

a) The mappings $\varphi$ and $\varphi '$ are homeomorphisms;

b) The joint spectrum Sp$^{\Lambda}_B((x_{\lambda}))$ is the polynomially convex envelope of Sp$^{\Lambda}_{\mathscr{C}(X)}((x_{\lambda}))$;

c) The mapping $\varphi$ transforms $\mathscr{C}(X)$ into $\mathscr{C}$(Sp$^{\Lambda}_{\mathscr{C}(X)}((x_{\lambda}))$) and B into P(Sp$^{\Lambda}_{\mathscr{C}(X)}((x_{\lambda}))$) ;

d) The mapping $\varphi '$ transforms $\mathscr{G}_B(B)$ into P(Sp$^{\Lambda}_B((x_{\lambda}))$) ;

e) The mappings $\varphi$ and $\varphi '$ transform $\mathscr{G}_B$ into the canonical isomorphism of P(Sp$^{\Lambda}_{\mathscr{C}(X)}((x_{\lambda}))$) onto P(Sp$^{\Lambda}_B((x_{\lambda}))$).

The mappings $\varphi$ and $\varphi '$ are continuous and surjective (No.$^o6$ of I, p. 41). The mapping $\varphi '$ is a homeomorphism by assertion a) of Prop. 12 of I, p. 43, and $i$ is injective, hence $\varphi$ is injective. Therefore $\varphi$ and $\varphi '$ are homeomorphisms.

Let $X_1$ = Sp$^{\Lambda}_{\mathscr{C}(X)}((x_{\lambda}))$. The polynomially convex envelope of $X_1$ is $Y_1=$ Sp$^{\Lambda}_B((x_{\lambda}))$ by Prop. 14 of I, p. 46.

For every $\lambda \in \Lambda$, denote by $z_{\lambda}$ the corresponding coordinate function on $\mathbf{C}^{\Lambda}$. The mapping $\varphi$ transforms $x_{\lambda}$ into $z_{\lambda}|X_1$, and $\varphi '$ transforms $\mathscr{G}_B(x_{\lambda})$ into $z_{\lambda}|Y_1$. Hence $\varphi$ transforms B into $P(X_1)$, and $\varphi '$ transforms $\mathscr{G}_B(B)$ into $P(Y_1)$. Finally, $\varphi$ and $\varphi '$ transform $\mathscr{G}_B$ into the canonical isomorphism of $P(X_1)$ onto $P(Y_1)$.