Galois Correspondence for Free Actions of Compact Abelian Groups on C^*-Algebras and Generic Points of Invariant Measures

Abstract

We deal with algebras of sphericl functions associated with covariant systems over a compact group with locally compact group action on 𝐶∗-algebras and compact subgroups and duality theory for nonergodic actions. The quasi product actions of compact abelian group on a 𝐶∗ - algebra and freeness of actions of finite abelian groups on 𝐶∗-algebras and free of compact quantum groups on unital 𝐶∗ - algebras are considered. The Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, for compact quantum group actions and 𝐶∗-algebras are studied. The homoclinic groups and expansive algebraic actions are presented. The invariant measures for homeomorphisms with weak specification and orbit equivalence for generalized Toeplitz subshifts are introduced. The generic points of invariant measures for an amenable residually finite group actions with the weak specification property for ergodic group automorphisms of abelian groups are characterized.