Spectral decompositions of spaces induced by spectral decompositions of acting groups

The presence on a space of an additional algebraic structure, which is compatible with its topology, in many cases imposes strong restrictions on the properties of the space itself. In the paper we discuss connections between spectral characterizations of groups and spaces with their actions when the latter are near to their coset spaces. The compatible system of maps on a space which is induced by a compatible system of maps on the acting group is constructed. This approach allows to give a unified possibility in the study of spaces with special actions of groups with suitable spectral decompositions. Among them are Čech complete groups, their products and subgroups. In particular, it is proved that a pseudocompact coset space of a Čech-complete group is ϰ-metrizable and a compactum which is a coset space of a subgroup of a product of Čech-complete groups is an openly generated compactum. We also show that an arbitrary d-open action of an ℵ0-balanced group on a pseudocompact space can be replaced by a d-open action of an ℵ0-bounded group.