# Extensions of MF algebras and Volume Entropy in Finite von Neumann Algebras

#### Abstract

In this dissertation, we give conditions for an extension of an MF algebra to be MF. Initially we consider the C*-algebra generated by the image of an MF algebra under a faithful and essential *-representation and compact operators and show that such an algebra will be MF. K-theory, KK theory and the Universal Coefficient Theorem are the main tools utilized to extend these results to more general extensions extensions. MF algebras are closely connected to Voiculescu's notion of free entropy in C*-algebras and this leads naturally to the second chapter, dealing with the calculation of volume entropy. The second chapter of this dissertation is concerned with volume entropy in finite von Neumann algebras. These concepts fall within the field of free probability. Free probability is an extension of probability to the setting of von Neumann algebras. This generalized notion of probability has seen a large degree of success in studying the structure and properties of von Neumann algebras. In the second chapter we consider volume entropy of n-tuples of elements in a finite von Neumann algebra and prove an analogue of Szego's Limit Theorem.

*This paper has been withdrawn.*