On extension of the group operation over the Čech-Stone compactification

The convolution of ultrafilters of closed subsets of a normal topological group is considered as a substitute of the extension onto ${\displaystyle {(\beta )}^2}$ of the group operation. We find a subclass of ultrafilters for which this extension is well-defined and give some examples of pathologies. Next, for a given locally compact group and its dense subgroup , we construct subsets of β algebraically isomorphic to . Finally, we check whether the natural mapping from β onto β is a homomorphism with respect to the extension of the group operation. All the results involve the existence of R-points.