Strongly invariant means on commutative hypergroups
Volume 129 / 2012
Colloquium Mathematicum 129 (2012), 119-131
MSC: Primary 43A62; Secondary 43A07.
DOI: 10.4064/cm129-1-9
Abstract
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.
