Tracial Rokhlin Property and Non-Commutative Dimensions

Abstract

This dissertation focuses on finite group actions with the tracial Rokhlin property and the structure of the corresponding crossed products. It consists of two major parts. For the first part, we study several different aspects of finite group actions with certain versions of the Rokhlin property. We are able to give an explicit characterization of product-type actions with the tracial Rokhlin property or strict Rokhlin property. We also show that, in good circumstances, the actions with the tracial Rokhlin property are generic. In the second portion of this dissertation, we introduce the weak tracial Rokhlin property for actions on non-simple C*-algebras. The main results are as follows. Let A be a unital non-simple C*-algebra and α be an action of G on A with the weak tracial Rokhlin property. Assume that the crossed product C*: G, A, α) is simple. Suppose A has either of the following property: tracial rankk, stable rank one, real rank zero. Then C*: G, A, α) has the same property.