Takai duality and crossed product hosts in C*-actions

Crossed product algebras are fundamental in the study of C*-algebras, traditionally under the assumption of continuity of group actions. Recent work by Grundling and Neeb introduced the crossed product host, an analog of the crossed product for a singular action. In this paper, we investigate the structure of the crossed product host and its relation to the conventional crossed product. We examine the validity of Takai-type duality in this setting and establish connections via the Landstad algebra. Additionally, we provide a necessary and sufficient condition for the existence of ground states, and show that for amenable groups, if the full and reduced crossed product hosts exist, then they coincide.