  Our main result is to give necessary and sufficient conditions, in terms of
Fourier transforms, on a closed ideal $I$ in $\loneg$, the space of radial
integrable functions on $G=SU(1,1)$, so that $I=\loneg$ or $I=\lonez$---the
ideal of $\loneg$ functions whose integral is zero. This is then used to prove
a generalization of Furstenberg's theorem which characterizes harmonic
functions on the unit disk by a mean value property and a ``two circles" Morera
type theorem (earlier announced by Agranovski\u{\i}).
