Harmonic projection and hypercentral extensions

TL;DR

This paper investigates the preservation of the Liouville property under hypercentral extensions, introducing a harmonic projection to analyze its behavior in group extensions.

Contribution

It demonstrates that the Liouville property can be preserved in certain hypercentral extensions using a new harmonic projection method.

Findings

Liouville property preserved in some hypercentral extensions

Introduction of a projection from ℓ∞ to harmonic functions

Provides insights into the structure of amenability and extensions

Abstract

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 $\ell^\infty$ onto the space of harmonic functions is introduced.