Abstract

This dissertation consists of two related parts. In the first portion we use the tracial Rokhlin property for actions of a finite group G on stably finite simple unital C*-algebras containing enough projections. The main results of this part of the dissertation are as follows. Let A be a stably finite simple unital C*-algebra and suppose a is an action of a finite group G with the tracial Rokhlin property. Suppose A has real rank zero, stable rank one, and suppose the order on projections over A is determined by traces. Then the crossed product algebra C* (G, A, α) also has these three properties.

In the second portion of the dissertation we introduce an analogue of the tracial Rokhlin property for C*-algebras which may not have any nontrivial projections called the projection free tracial Rokhlin property. Using this we show that under certain conditions if A is an infinite dimensional simple unital C*-algebra with stable rank one and α is an action of a finite group G with the projection free tracial Rokhlin property, then C* (G, A, α) also has stable rank one.