Wednesday, November 2nd, 2011
2:00pm - 3:00pm, in Science 2-064

Jonathan Brown

University of Otago (New Zealand)

Permanence properties of crossed products associated to Rokhlin actions

Abstract: $C^*$-algebras are complete $*$-algebras which generalize the algebra of $n$ by $n$ matrices over the complex numbers. They are used (among other things) to study group representations and group actions. The $C^*$-algebras used in these applications are constructed from a group $G$ acting by automorphisms on another $C^*$-algebra $A$. We call the resulting algebra the crossed product of $A$ by $G$. We are interested in the properties of $A$ that are shared by the crossed product of $A$ by $G$. For instance, a $C^*$-algebra is AF if it can be approximated by finite dimensional matrix algebras. AF-algebras are an important class of $C^*$-algebras that are both useful in applications and well understood. Given a finite group $G$ acting on an AF-algebra $A$, we show, using the work of N. Christopher Phillips, that if the action of $G$ on $A$ satisfies a certain ``freeness'' condition called the Rokhlin property, then the crossed product is also AF.