Date of Award
Program or Major
Doctor of Philosophy
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.
Wu, Jinsong, "On decompositions and Connes's embedding problem of finite von Neumann algebras" (2011). Doctoral Dissertations. 630.