An F. and M. Riesz theorem for bounded symmetric domains (Q1085379)

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 metrizable 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.