An F. and M. Riesz theorem for bounded symmetric domains

Brummelhuis, R. G. M.

Annales de l'Institut Fourier, Tome 37 (1987) no. 2, pp. 139-150.

Le théorème de F. et M. Riesz classique est étendu aux groupes compacts et métrisables dont le centre contient une copie du groupe du cercle. Des exemples importants de tels groupes sont les groupes d’isotropie des domaines symétriques bornés.

La preuve se sert d’un critère pour la continuité absolue qui emploie les espaces $L^p$ avec $p<1$ : une mesure $\mu$ sur un groupe $K$ métrisable et compact est absolument continue par rapport à la mesure de Haar $dk$ de $K$ si, pour un $p<1$, un certain sous-espace de $L^p(K,dk)$, dépendant de $\mu$, a un nombre suffisant de fonctionnelles linéaires continues pour séparer les points. Si $K$ est abélien ce critère est dû à J.H. Shapiro.

We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains.

The proof uses a criterion for absolute continuity involving $L^p$ spaces with $p<1$: A measure $\mu$ on a compact metrisable group $K$ is absolutely continuous with respect to Haar measure $dk$ on $K$ if for some $p<1$ a certain subspace of $L^p(K,dk)$ which is related to $\mu$ has sufficiently many continuous linear functionals to separate its points. For abelian $K$ this criterion is due to J.H. Shapiro.

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