Exercise 8, § 8

Let X be a compact topological space and B a unital Banach subalgebra of $\mathscr{C}(X)$ (for the induced norm), which separates the points of X. We identify X with a closed subset of $\mathsf{X}(B) = X'$. Let $B^*$ be the set of invertible elements of B. One says that B is logmodular if the set of functions log($|f|$), for $f\in B^*$, is dense in $\mathscr{C}_{\mathbf{R}}(X)$. (This is the case, in particular, if the set of functions $\mathscr{R}(f)$, for $f\in B$, is dense in $\mathscr{C}_{\mathbf{R}}(X)$, since log($|e^f|$) $=\mathscr{R}(f)$.) In the sequel, we suppose that B is logmodular.

a) For every $\chi \in X'$, there exists a unique measure $\mu_{\chi}\geqslant 0$ on X such that

log($|\chi (f)|$) $=\int_X$ log($|f(\omega )|$)$d\mu_{\chi}(\omega )$ for every $f\in B^*$, hence such that $\chi (f) =\int_Xf(\omega )d\mu_{\chi}(\omega )$ for every $f\in B$.

b) One has

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

for every $f\in B$ and the condition $\chi (f) =\int_Xf(\omega )d\mu_{\chi}(\omega )$ for every $f\in B$ determines $\mu_{\chi}$ uniquely.

c) Let $\chi \in X'$ and $\mu$ be a positive measure on X. Let us write $\mu=\mu_1+\mu_2$ where $\mu_1$ has base $\mu_{\chi}$, and where $\mu_2$ and $\mu_{\chi}$ are mutually singular. Let I be the kernel of $\chi$. Then

inf$_{f\in}\int_{IX}|1-f|^2d\mu=$ inf$_{f\in}\int_{IX}|1-f|^2d\mu_1$.

d) Let $\chi \in X'$ and $h$ be a function $\geqslant 0$ of $\mathscr{L}^1(\mu_{\chi})$. Let I be the kernel of $\chi$. Then

inf$_{f\in I}\int_X|1-f|^2h d\mu_{\chi}=$ exp$(\int_X$ log($h$)$d\mu_{\chi})$.

e) For every positive measure $\mu$ on X, and every $p\in [1,+\infty [$, let $H^p(\mu)$ be the closure of the canonical image of B in $L^p(\mu)$. Let $\chi \in X'$ and I the kernel of $\chi$. Then $L^2(\mu_{\chi})$ is the Hilbert sum of $H^2(\mu_{\chi})$ and of the closure in $L^2(\mu_{\chi})$ of the canonical image of $\overline{I}$ (the set of conjugates of the functions belonging to I).

f) The space $H^1(\mu_{\chi})$ is the set of classes $\widetilde{h}$ of the $h\in \mathscr{L}^1(\mu_{\chi})$ such that $\int_Xf h d\mu_{\chi}= 0$ for every $f\in I$.

g) Let $h$ be a function $\geqslant 0$ in $\mathscr{L}^1(\mu_{\chi})$. Then one has $\widetilde{h}=|\widetilde{f}|$ with $\widetilde{f}\in H^1(\mu_{\chi})$ and $\int_Xf d\mu_{\chi}\not= 0$ if and only if log($h$)$\in \mathscr{L}^1(\mu_{\chi})$.