Tracial Rokhlin property for actions of amenable groups

Tracial Rokhlin property was introduced by Chris Phillips to study the structure of crossed product of actions on simple C*-algebras. It was originally defined for actions of finite groups and group of integers. Matui and Sato generalized it to actions of amenable groups. In this paper, we give a further generalization of Matui and Sato's definition. We shall show that many known results about tracial Rokhlin property could be generalized to actions of amenable groups under this definition, and some natural examples has the (weak) tracial Rokhlin property, at least for a special class of groups called ST-tileable groups.