Strongly invariant means on commutative hypergroups

Rupert Lasser ¹   ; Josef Obermaier ²

¹ Helmholtz National Research Center for Environment and Health Institute of Biomathematics and Biometry Ingolstädter Landstraße 1 85764 Neuherberg, Germany and Munich University of Technology Centre of Mathematics 85748 Garching, Germany
² Helmholtz National Research Center for Environment and Health Institute of Biomathematics and Biometry Ingolstädter Landstraße 1 85764 Neuherberg, Germany

Colloquium Mathematicum, Tome 129 (2012) no. 1, pp. 119-131

Résumé

We introduce and study strongly invariant means $m$ on commutative hypergroups, $m(T_x\varphi \cdot \psi)=m(\varphi \cdot T_{\tilde{x}}\psi)$, $x \in K$, $\varphi,\psi \in L^\infty(K)$. We show that the existence of such means is equivalent to a strong Reiter condition. For polynomial hypergroups we derive a growth condition for the Haar weights which is equivalent to the existence of strongly invariant means. We apply this characterization to show that there are commutative hypergroups which do not possess strongly invariant means.

Détail

DOI : 10.4064/cm129-1-9

Keywords: introduce study strongly invariant means commutative hypergroups varphi cdot psi varphi cdot tilde psi varphi psi infty existence means equivalent strong reiter condition polynomial hypergroups derive growth condition haar weights which equivalent existence strongly invariant means apply characterization there commutative hypergroups which possess strongly invariant means