Free probability, free entropy and generator problems for von Neumann algebras

The study of von Neumann algebras was initiated by Murray and von Neumann in the thirties of the last century. They separated the family of von Neumann algebras into three types, I, II, and III. There have been extensive developments following their fundamental work. Our focus is on the finite, type II von Neumann algebras. In the early 1980s, D. Voiculescu began the development of the theory of free probability and free entropy. This new tool was crucial in solving some old open problems in the field of type II1 von Neumann algebras. The first breakthrough was made by Voiculescu when he showed that each free group factor L(F( n)), (n ≥ 2), has no Cartan subalgebra. Later, Ge showed that each free group factor L(F( n)), (n ≥ 2), is prime. The first part of this thesis is devoted to proving the Theorem 2.5.1, which is used to compute the free entropy of an important class of II 1 von Neumann algebras, L(SL([special characters omitted], 2n + 1)) (n ≥ 2). The theorem also gives the generalization of the results of Voiculescu and Ge that were mentioned as above. The second part of this thesis is on the generator problems for von Neumann algebras. We give an affirmative answer to Voiculescu\u27s question on the generator problem for the group von Neumann algebras associated with special linear groups with integer entries