Free entropy dimensions and approximate liftings

In the first chapter of the dissertation, we give a very elementary proof of a more detailed version of one of D. Voiculescu\u27s results, which was a key ingredient in Voiculescu\u27s proof that his free entropy is additive when the variables are free. In the second chapter of the dissertation, based on the notion of upper free orbit-dimension introduced by D. Hadwin and J. Shen, we introduce a new invariant on finite von Neumann algebras that do not necessarily act on separable Hilbert space. We show that this invariant is independent of the generating set, and we obtain a number of results for von Neumann algebras that are not finitely generated. In the third chapter of the dissertation, we consider the class of approximately divisible C*-algebras. Let A be a separable unital approximately divisible C*-algebra. We show that A is generated by two self-adjoint elements and the topological free entropy dimension of any finite generating set of A is less than or equal to 1. In addition, we show that the similarity degree of A is at most 5. In the fourth chapter of the dissertation, we show that two (weakly) semiprojective unital C*-algebras, each generated by n projections, can be glued together with partial isometries to define a larger (weakly) semiprojective algebra. In the von Neumann algebra setting, we prove lifting theorems for trace-preserving *-homomorphisms from abelian von Neumann algebras or hyperfinite von Neumann algebras into ultraproducts. We also extend and simplify a classical result of S. Sakai by showing that a tracial ultraproduct of C*-algebras is a von Neumann algebra