Harmonic projection and hypercentral extensions

The Liouville property is a strong form of amenability, but contrary to amenability, it is not well-behaved under extensions. In this paper it is shown that, for some measures, the Liouville property is preserved by [FC-]hypercentral extensions. To this end a projection from ${\mathrm{\ell }}^{\mathrm{\infty }}$ onto the space of harmonic functions is introduced.