Proposition 12

Let Λ be a set and A a commutative unital Banach algebra. Let $x= (x_{\lambda})_{\lambda\in\Lambda}$ be a family of elements of A.

a) Suppose that the full subalgebra of A generated by the family $x$ is dense in A. The mapping of $\mathsf{X}(A)$ into $\mathbf{C}^{\Lambda}$ defined by $x$ is a homeomorphism of $\mathsf{X}(A)$ onto the joint spectrum Sp$^{\Lambda}_A(x)$ ;

b) Suppose that the unital subalgebra of A generated by the family $x$ is dense in A. For every $c\in \mathbf{C}^{\Lambda}$, the following conditions are equivalent :

(i) $c\in$ Sp$^{\Lambda}_A(x)$ ;

(ii) $|P(c)|\leqslant \varrho (P(x))$ for every polynomial $P\in \mathbf{C}[(X_{\lambda})_{\lambda\in\Lambda}]$;

(iii) $|P(c)|\leqslant \|P(x)\|$ for every polynomial $P\in \mathbf{C}[(X_{\lambda})_{\lambda\in\Lambda}]$.

a) The mapping of $\mathsf{X}(A)$ into Sp$^{\Lambda}_A(x)$ defined by the family $x$ is continuous and surjective. Let $\chi ,\chi '\in \mathsf{X}(A)$ be characters having the same image, that is, such that $\chi (x_{\lambda}) =\chi '(x_{\lambda})$ for every $\lambda \in \Lambda$. The characters $\chi$ and $\chi '$ coincide on elements of the form $P(x)Q(x)^{-1}$, where $P\in \mathbf{C}[(X_{\lambda})]$, $Q\in \mathbf{C}[(X_{\lambda})]$ and $Q(x)$ is invertible in A, that is, on the full subalgebra of A generated by the elements $x_{\lambda}$ (lemma 2 of I, p. 6). Since $\chi$ and $\chi '$ are continuous (Theorem 1 of I, p. 29), they are therefore equal on A. This proves that $\mathsf{X}(h)$ is a continuous bijection of $\mathsf{X}(A)$ onto Sp$^{\Lambda}_A(x)$, and hence is a homeomorphism since $\mathsf{X}(A)$ is compact.

b) Let us show that (i) implies (ii): if $c= (c_{\lambda})_{\lambda\in\Lambda}\in$ Sp$^{\Lambda}_A(x)$, there exists $\chi \in \mathsf{X}(A)$ such that $c_{\lambda}=\chi (x_{\lambda})$ for every $\lambda$. For every $P\in \mathbf{C}[(X_{\lambda})]$, we therefore have $|P(c)|=|P((\chi (x_{\lambda}))_{\lambda\in\Lambda})|=|\chi (P(x))|\leqslant \varrho (P(x))$.

Assertion (ii) implies (iii) by reason of the inequality $\varrho (x)\leqslant \|x\|$, valid for every $x\in A$ by definition of the spectral radius.

Finally let us show that (iii) implies (i). Let $c= (c_{\lambda})_{\lambda\in\Lambda}\in \mathbf{C}^{\Lambda}$ be such that

for every P $\in \mathbf{C}[(X_{\lambda})]$. Let $A'$ be the unital subalgebra of A generated by the family $x$; its elements are of the form $P(x)$ for $P\in \mathbf{C}[(X_{\lambda})]$. The majorisation (2) implies that the condition $P(x) = 0$ entails $P(c) = 0$. There therefore exists a morphism of unital algebras $\xi$ from $A'$ into $\mathbf{C}$ such that $\xi (x_{\lambda}) =c_{\lambda}$ for every $\lambda \in \Lambda$. By (2), the morphism $\xi$ is continuous. It therefore extends by continuity to a character $\chi$ of $\overline{A'}= A$, which satisfies $c= (\chi (x_{\lambda}))_{\lambda\in\Lambda}\in$ Sp$^{\Lambda}_A(x)$. This completes the proof.