On mixing and ergodicity in locally compact motion groups

Abstract

Let G be a semi-direct product with A Abelian and K compact. We characterize spread-out probability measures on G that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral radius formula for the Fourier transform of a regular Borel measure on G that we develop, and which is analogous to the well-known Beurling-Gelfand spectral radius formula. For spread-out probability measures on G, we also characterize ergodicity by convolutions by means of the Fourier transform of the measure. Finally, we show that spread-out probability measures on such groups are mixing by convolutions if and only if they are weakly mixing by convolutions.