Proposition 3

Let X be a compact space. Let B be a unital Banach subalgebra of $\mathscr{C}(X)$, endowed with the induced norm, separating the points of X. We identify X with a closed subset of $\mathsf{X}(B)$ (prop. 1, d)).

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.