Adhérence faible étoile d'algèbres de fractions rationnelles

Chaumat, Jacques

Annales de l'Institut Fourier, Tome 24 (1974) no. 4, pp. 93-120.

Étant donnés un compact $K$ du plan complexe, et une mesure non nulle sur $K$, on étudie $H^{\infty }\left(\mu \right)$, l’adhérence dans $L^{\infty }\left(\mu \right)$, pour la topologie $\sigma \left(L^{\infty }\right(\mu ),L^1(\mu \left)\right)$, de l’algèbre des fractions rationnelles d’une variable complexe, à pôles hors de $K$. Le résultat principal obtenu est qu’il existe un sous-ensemble $E_{\mu }$ de $K$, éventuellement vide, mesurable pour la mesure de Lebesgue plane, et une mesure ${\mu }_s$, éventuellement nulle, absolument continue par rapport à la mesure $\mu$, tels que : $H^{\infty }\left(\mu \right)$ soit isométriquement isomorphe à $H^{\infty }({\lambda }_{E_{\mu }})\oplus L^{\infty }({\mu }_s)$, où ${\lambda }_{E_{\mu }}$ désigne la restriction à $E_{\mu }$ de la mesure de Lebesgue plane.

Let $K$ be a compact subset of the complex plane, and $\mu$ a measure on $K$; we study $H^{\infty }\left(\mu \right)$, the weak star closure in $L^{\infty }\left(\mu \right)$, of the algebra of rational functions with poles off $K$. The main result is the following: there exists a subset $E_{\mu }$ of $K$, possibly empty, measurable with respect to the Lebesgue measure, and a measure ${\mu }_s$, possibly equal to zero, absolutely continuous with respect to the measure $\mu$, such that: $H^{\infty }\left(\mu \right)$ is isometrically isomorphic to $H^{\infty }({\lambda }_{E_{\mu }})\oplus L^{\infty }({\mu }_s)$, with ${\lambda }_{E_{\mu }}$ the restriction to $E_{\mu }$ of the Lebesgue measure.

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