Uniformly cyclic vectors
Volume 104 / 2006
Colloquium Mathematicum 104 (2006), 21-32
MSC: 43A65, 43A15, 43A77, 43A07.
DOI: 10.4064/cm104-1-2
Abstract
A group acting on a measure space $(X,\beta,\lambda)$ may or may not admit a cyclic vector in $L_\infty(X)$. This can occur when the acting group is as big as the group of all measure-preserving transformations. But it does not occur, even though there is no cardinality obstruction to it, for the regular action of a group on itself. The connection of cyclic vectors to the uniqueness of invariant means is also discussed.
