a) The mapping $j$ has norm $\leqslant 1$ in $\mathscr{L}(B;\mathscr{C}(X))$ ;

b) The radical of the algebra B is zero ;

c) The mapping ev is continuous from X into $\mathsf{X}(B)$ ;

d) If B separates the points of X, then the mapping ev is a homeomorphism of X onto a closed subset of $\mathsf{X}(B)$.

One can identify X with $\mathsf{X}(\mathscr{C}(X))$ and $\mathscr{G}_{\mathscr{C}(X)}$ with the identity mapping (example 1 of I, p. 36). The mapping ev is then identified with $\mathsf{X}(j)$, which proves c).

For every function $f\in B$ and every $x\in X$, one has $\mathscr{G}_B(f$)(ev$_x$) $=f(x)$, whence

$\|f\|_{\mathscr{C}(X)}=$ sup$_{x\in X}|\mathscr{G}_B(f$)(ev$_x$)$|\leqslant \varrho_B(f)\leqslant \|f\|_B$,

(cf. prop. 7 of I, p. 38, a)) which entails a). Moreover, this shows that the Gelfand transform of B is injective, whence b) (prop. 8 of I, p. 38). If B separates the points of X, then the mapping ev is injective, whence d) since X is compact.

We now consider the surjectivity of the mapping ev.

a) If the mapping ev is surjective, then the algebra B is a full subalgebra of $\mathscr{C}(X)$ ;

b) Suppose that B is a full subalgebra of $\mathscr{C}(X)$, and that there exists an element $a\in B$ such that the set of elements $f(a)$, where $f$ ranges over the set of rational functions on $\mathbf{C}$ without poles on Sp$_B(a)$, is dense in B. Then the mapping ev is surjective ;

c) If B is a full involutive subalgebra of $\mathscr{C}(X)$, then the mapping ev is surjective.

Assertion a) follows from prop. 10 of I, p. 40, and assertion b) from prop. 11 of I, p. 41, since the full subalgebra of B generated by $a$ is the set of elements $f(a)$, where $f$ ranges over the set of rational functions on $\mathbf{C}$ without poles on Sp$_B(a)$ (lemma I, p. 6 and prop. 6 of I, p. 37, b)).

Let us prove c). Suppose then that B is a full involutive subalgebra of $\mathscr{C}(X)$. To prove that ev is surjective, it is enough to prove that for every $\chi \in \mathsf{X}(B)$, there exists $y\in X$ such that Ker($\chi$ ) $=$ Ker(ev$_y$) (th. 2 of I, p. 30). Let I = Ker($\chi$ ). It is a maximal ideal of B. Let Φ be the set of $x\in X$ such that $f(x) = 0$ for every $f\in I$. Let us prove that Φ is not empty. Otherwise, since X is compact, there would exist an integer $n\geqslant 1$, an open covering $(V_1, . . . ,V_n)$ of X and, for every integer $i$ such that $1\leqslant i\leqslant n$, a function $f_i\in I$ such that $f_i(x)\not= 0$ for every $x\in V_i$. Since the algebra B is an involutive subalgebra of $\mathscr{C}(X)$, the function

would belong to I. Now $f(x)>0$ for every $x\in X$, and therefore $f$ would be invertible in $\mathscr{C}(X)$. Since B is supposed to be a full subalgebra of $\mathscr{C}(X)$, the function $f\in$ I would be invertible in B, which is impossible. Consequently, the set Φ is not empty. Let $y$ be an element of Φ; the kernel of the character ev$_y$ contains I, and is therefore equal to I.

a) For every $f\in B$, the Gelfand transform $\mathscr{G}_B(f)$ is a continuous function on $\mathsf{X}(B)$ which extends $f$ and satisfies $\|f\|=$ sup$|\mathscr{G}_B(f)|$. In particular, $\mathscr{G}_B$ is an isometric isomorphism of B onto a Banach subalgebra of $\mathscr{C}(\mathsf{X}(B))$ ;

b) Let $B^*$ be the set of invertible elements of B. For every character $\chi \in \mathsf{X}(B)$, there exists a positive measure $\mu$ of mass 1 on X such that, for every $f\in B^*$, one has

log($|\chi (f)|$) $=\int_X$ log($|f|$)$d\mu$.

Moreover, for every $f\in B$, one has

c) Suppose that every element of $\mathscr{C}_{\mathbf{R}}(X)$ is the uniform limit of real parts of functions belonging to B. Then, for every $\chi \in \mathsf{X}(B)$, there exists a unique measure $\mu_{\chi}\geqslant 0$ on X such that, for every function $f\in B$, one has

Moreover, for every function $f\in B$, one has

log($|\chi (f)|$)$\leqslant \int_X$ log($|f|$)$d\mu_{\chi}$; the function log$|f|$ being bounded above, the integral exists in $\mathbf{R}\cup \{-\infty \}$.

Assertion a) follows from the identification of X with a closed subspace of $\mathsf{X}(B)$ and from the inequalities

$\|f\|=$ sup$_{x\in X}|f(x)|\leqslant$ sup$_{\chi\in\mathsf{X}(B)}|\chi (f)|=$ sup$|\mathscr{G}_B(f)|=\varrho_B(f)\leqslant \|f\|$

for every $f\in B$.

Let $\chi \in \mathsf{X}(B)$ and $n$ be a positive integer. Let $\lambda_1, . . . , \lambda_n\in \mathbf{R}$ and $f_1, . . . , f_n\in B^*$. One then has

(1) $\sum_{i=1}^n\lambda_i$ log($|\chi (f_i)|$)$\leqslant$ sup$_{x\in X}(\sum_{i=1}^n\lambda_i$ log($|f_i(x)|$)).

Indeed, by continuity, it is enough to prove this inequality when the real numbers $\lambda_i$ are rational. Reducing to a common denominator, one is reduced to the case where $\lambda_i\in \mathbf{Z}$ for every $i$. The inequality then takes the form

log($|\chi (f_1^{\lambda_1}\cdots f_n^{\lambda_n})|$)$\leqslant$ sup$_{x\in X}$ log($|(f_1^{\lambda_1}\cdots f_n^{\lambda_n})(x)|$),

and follows from the fact that $\|\chi \|= 1$ (th. 1 of I, p. 29).

Let $B'$ be the vector space generated in $\mathscr{C}_{\mathbf{R}}(X)$ by the functions log($|f|$) for $f\in B^*$. The upper bound (1) proves that there exists a linear form $h$ of norm $\leqslant$ 1 on $B'$ such that log($|\chi (f)|$) $=h$(log($|f|$)) for every $f\in B^*$. By the Hahn-Banach theorem (EVT, II, p. 24, cor. 2), the linear form $h$ extends to a linear form $\mu$ of norm $\leqslant 1$ on $\mathscr{C}_{\mathbf{R}}(X)$, that is, to a real measure $\mu$ on X such that $\|\mu\|\leqslant 1$ (INT, III, §1, n$^o5$). Taking for the element $f$ of $B^*$ the constant $e=$ exp(1), one sees that $1 =\mu(1)$. Hence, writing $\mu=\mu^+-\mu^-$ as the difference of two mutually singular positive measures (INT, III, §1, n$^o6$, th. 3), one obtains

whence $\mu=\mu^+\geqslant 0$ and $\|\mu\|= 1$.

For every $f\in B$, we have exp($f$)$\in B^*$, hence

$\int_X\mathscr{R}(f)d\mu=\int_X$ log($|$exp($f$)$|$)$d\mu=$ log($|\chi$(exp($f$))$|$)

= log($|$exp($\chi (f)$)$|$) $=\mathscr{R}(\chi (f))$,

where Cor. 1 of I, p. 66 has been used. Applying this equality to $if$, we conclude that $\int_Xf d\mu=\chi (f)$. This establishes b).

Let us place ourselves under the assumptions of c). The existence of $\mu_{\chi}$ follows from b). On the other hand, we have $\mu_{\chi}(\mathscr{R}(f)) =\mathscr{R}(\chi (f))$ for every $f\in B$. Since the real parts of functions $f\in B$ are dense in $\mathscr{C}_{\mathbf{R}}(X)$ by hypothesis, the measure $\mu_{\chi}$ is uniquely determined by $\chi$.

Let $f\in B$. Let $\varepsilon >0$ be a real number. There exists, by hypothesis, a function $g\in B$ such that

(2) $\mathscr{R}(g)-\varepsilon \leqslant$ log($|f|+\varepsilon$ )$\leqslant \mathscr{R}(g) +\varepsilon$.

Let $h=$ exp($g$)$\in B^*$. By (2), we have

The upper bound implies $|f h^{-1}|\leqslant e^{\varepsilon}$, whence $|\chi (f h^{-1})|\leqslant e^{\varepsilon}$, and consequently

(4) log($|\chi (f)|$)$\leqslant$ log($|\chi (h)|$) $+\varepsilon =\int_X$ log($|h|$)$d\mu_{\chi}+\varepsilon$.

The lower bound in (3) then implies

log($|\chi (f)|$)$\leqslant \int_X$ log $(|f|+\varepsilon )d\mu_{\chi}+ 2\varepsilon$.

Letting $\varepsilon$ tend to 0, we deduce that

log($|\chi (f)|$)$\leqslant \int_X$ log($|f|$)$d\mu_{\chi}$.

This completes the proof.

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 $P(Y_1)$.

a) The mapping $\theta$ from $\mathsf{X}(P(X))$ onto Sp$^{\Lambda}_{P(X)}((z_{\lambda}))$ defined by the family $(z_{\lambda})$ is a homeomorphism of $\mathsf{X}(P(X))$ onto the polynomially convex envelope Y of X. Its restriction to X is the identity mapping of X ;

b) For every function $f\in P(X)$, the homeomorphism $\theta$ transforms the prolongation $\mathscr{G}_{P(X)}(f)$ of $f$ to $\mathsf{X}(P(X))$ into a prolongation $\widetilde{f}$ of $f$ to Y. The mapping $f\mapsto \widetilde{f}$ is the canonical isomorphism of P(X) onto P(Y).

In prop. 4, take B = P(X) and $x_{\lambda}=z_{\lambda}$. Then $\varphi$ becomes the identity mapping and $\varphi '$ becomes the mapping $\theta$. The assertions of the corollary are then reduced to those of loc. cit.

If X is connected, the only idempotents of $\mathscr{C}(X)$, and hence of P(X), are 0 and 1 (cor. of prop. I, p. 79). Consequently, the space $\mathsf{X}(P(X))$ is connected (loc. cit.); now this set is homeomorphic to the polynomially convex envelope of X (cor. 1, a)).

The closure of the set O in $\mathbf{C}-$ X is equal to $\overline{O}\cap (\mathbf{C}-$ X) where $\overline{O}$ is its closure in $\mathbf{C}$. Since O is a connected component of $\mathbf{C}-$ X, we therefore have $\overline{O}\cap (\mathbf{C}-$ X) = O, which proves that $\overline{O}-O\subset X$.

Let X be a compact subset of $\mathbf{C}$. Let $O_{\infty}$ be the unbounded connected component of $\mathbf{C}-$X, and let $(O_i)_{i\in I}$ be the family of bounded connected components of $\mathbf{C}-$ X, the subsets $O_i$ being pairwise distinct.

Let E be a subset of $\mathbf{C}-$ X. We denote by $R_E(X)$ the closure in $\mathscr{C}(X)$ of the set of functions $f|X$, where $f$ is a rational function on $\mathbf{C}$ all of whose poles belong to E. The algebra $R_E(X)$ is a unital Banach subalgebra of $\mathscr{C}(X)$ which separates the points of X. Let $z$ be the identity function on X. The full closed subalgebra of $R_E(X)$ generated by $z$ is equal to $R_E(X)$ (Lemma 2 of I, p. 6). The elements of $R_E(X)$ are holomorphic in the interior of X.

In particular, one has $R_{\emptyset}(X) = P(X)$. We write $R(X) = R_{\mathbf{C}-X}(X)$. Let I(E) be the set of elements $i\in I$ such that $E\cap O_i=\emptyset$, and

The set $X_E$ is compact, being bounded and closed, its complement in $\mathbf{C}$ being the union of the open set $O_{\infty}$ and of the open sets $O_i$ which meet E.

a) The restriction mapping $h$ of $R_E(X_E)$ into $R_E(X)$ is an isometric isomorphism;

b) The algebra $R_E(X_E)$ is a full subalgebra of $\mathscr{C}(X_E)$;

c) Every character of $R_E(X_E)$ is of the form $f\mapsto f(w)$, where $w$ is an element of $X_E$;

d) The mapping $\chi \mapsto \chi (z)$ is a homeomorphism of $\mathsf{X}(R_E(X))$ onto $X_E$;

e) If $E'$ is a subset of $\mathbf{C}-$ X, one has $R_E(X) = R_{E'}(X)$ if and only if $X_E= X_{E'}$, which is also equivalent to $I(E) = I(E')$.

The restriction mapping $h$ of $R_E(X_E)$ into $R_E(X)$ is a morphism of Banach algebras such that $\|h(f)\|\leqslant \|f\|$ for every function $f\in R_E(X_E)$. Let $f\in R_E(X_E)$. Let $i\in I(E)$. Since $f$ is holomorphic in a neighbourhood of the bounded open set $O_i$, the maximum principle (VAR, I, p. 29) implies that there exists an element $z_0$ on the boundary of $O_i$ such that $|f(z_0)|=$ sup$_{z\in O_i}|f(z)|$. Since the boundary of $O_i$ is contained in X (lemma 1), it follows that sup$_{z\in O_i}|f(z)|\leqslant \|h(f)\|$. Since this inequality holds for every $i\in I(E)$, it follows that $\|f\|\leqslant \|h(f)\|$. The morphism $h$ is therefore isometric, and in particular injective. Let us now prove that it is surjective. Let $g\in R_E(X)$. There exists a sequence $(f_n)$ of rational functions whose poles belong to E which converges uniformly to $g$ on X. The $f_n|X_E$ are elements of $R_E(X_E)$. Since $h$ is isometric, the sequence $(f_n|X_E)$ converges in $\mathscr{C}(X_E)$. If $f$ is its limit, then $f\in R_E(X_E)$ and $g=f|X =h(f)$. Hence a).

Let us prove assertion d). Apply prop. 3 of I, p. 144 to the algebra $B = R_E(X)$. Assertion b) of loc. cit. implies that the mapping which to $\chi$ associates $\chi (z)$ is a homeomorphism of $\mathsf{X}(R_E(X))$ onto Sp$_{R_E(X)}(z)$. Let $z_E$ be the identical mapping of $X_E$. By loc. cit., a), we have Sp$_{R_E(X)}(z) =$ Sp$_{R_E(X_E)}(z_E)$. It is therefore enough to show that Sp$_{R_E(X_E)}(z_E) = X_E$. The cor. to prop. 6 of I, p. 28 proves that Sp$_{R_E(X_E)}(z_E)$ is the union of Sp$_{\mathscr{C}(X_E)}(z_E) = X_E$ and of certain bounded connected components of the complement of $X_E$. Let $O_i$ be one of the bounded connected components of the complement of $X_E$. The intersection $E\cap O_i$ is therefore nonempty. Let $\lambda \in E\cap O_i$; since $(\lambda -z_E)^{-1}\in R_E(X_E)$, we have $\lambda \notin$ Sp$_{R_E(X_E)}(z_E)$. Thus, $O_i$ is not contained in Sp$_{R_E(X_E)}(z_E)$. This shows that Sp$_{R_E(X_E)}(z_E) = X_E$.

Moreover, this equality implies that Sp$_{R_E(X_E)}(z_E) =$ Sp$_{\mathscr{C}(X_E)}(z_E)$. This establishes condition (iii) of prop. 11 of I, p. 41, applied to the canonical injection of $R_E(X_E)$ into $\mathscr{C}(X_E)$ and to the element $z_E$. Assertions b) and c) are the equivalent conditions (i) and (ii) of loc. cit.

Assertion d) shows that $X_E= X_{E'}$ if $R_E(X) = R_{E'}(X)$. Conversely, suppose that $X_E= X_{E'}$. Replacing $E'$ by $E\cup E'$, we may suppose that $E\subset E'$. By b), the algebra $R_E(X_E)$ is a closed full subalgebra of $\mathscr{C}(X_E)$, and hence also of $R_{E'}(X_E)$. It contains $z_E$, and therefore $R_E(X_E) = R_{E'}(X_E)$. Applying a), we deduce that $R_E(X) = R_{E'}(X)$. Finally, the equivalence of $X_E= X_{E'}$ and $I(E) = I(E')$ is a consequence of the definitions.

(i) The set E meets all the bounded connected components of $\mathbf{C}-$ X ;

(ii) The mapping $\chi \mapsto \chi (z)$ is a homeomorphism of $\mathsf{X}(R_E(X))$ onto X ;

(iii) One has $R_E(X) = R(X)$.

Let $E'=\mathbf{C}-$ X. The conditions (i), (ii) and (iii) are respectively equivalent to $I(E) = I(E')$, to $X_E= X_{E'}$ (by prop. 5, d)) and to $R_E(X) = R_{E'}(X)$. Hence they are equivalent among themselves by prop. 5, e).

The following corollary makes the theorem of Runge more precise (th. 3 of I, p. 69).

With $E =\{\lambda_i\}_{i\in I}$, the assumption is condition (i) of cor. 1. Hence $R_E(X) = R(X)$, and th. 3 of I, p. 69 shows that $R(X) =\mathscr{O}(X)$.