Through von Neumann's reduction theory, the classification of injective von Neumann algebras acting on separable Hilbert spaces translates into the classification of injective factors. In his proof of the uniqueness of the injective type II₁ factor, Connes showed an alternate characterization of the approximately inner automorphisms of type II₁ factors. Moreover, he conjectured that this characterization could be extended to all types of factors acting on separable Hilbert spaces. In this thesis, we present a general toolbox containing the basic notions needed to study von Neumann algebras, before describing our work concerning Connes' conjecture in the case of type IIIλ factors. We have obtained partial results towards the proof of a modified version of this conjecture.

This thesis arose from a desire to better understand the structures of automorphism groups and inner automorphism groups of quandles. We compute and give the structure of the automorphism groups of all dihedral quandles. In their paper Matrices and Finite Quandles, Ho and Nelson found all quandles (up to isomorphism) of orders 3, 4, and 5 and determined their automorphism groups. Here we find the automorphism groups of all quandles of orders 6 and 7. There are, up to isomoprhism, 73 quandles of order 6 and 289 quandles of order 7.