Date of Award

Fall 2011

Project Type


Program or Major


Degree Name

Doctor of Philosophy

First Advisor

Liming Ge


A longstanding open question of Connes asks whether every finite von Neumann algebra embeds into an ultraproduct of finite-dimensional matrix algebras. As of yet, algebras verified to satisfy Connes's embedding property belong to just a few special classes (e.g. amenable algebras and free group factors). In this dissertation we establish Connes's embedding property for von Neumann algebras satisfying Popa's co-amenability condition. Some decomposition properties of finite von Neumann algebras are also investigated.

Chapter 1 reviews von Neumann algebras, completely bounded mappings, conditional expectations, tensor products, crossed products, direct integrals, and Jones basic construction.

Chapter 2 introduces new decompositions of finite von Neumann algebras which we call F-thin, strongly F-thin, and weakly F-thin, etc. We also consider the singly-generated problem, and compute the cohomology in such decompositions of finite von Neumann algebras.

In Chapter 3 we show by estimation of free entropy that free group factors lack the type of decompositions discussed in Chapter 2.