This dissertation is devoted to several questions that arise in operator algebra theory. In the first part of the work we study the dilations of homomorphisms of subalgebras to the algebras that contain them. We consider the question whether a contractive homomorphism of a logmodular algebra into B(H) is completely contractive, where B(H) denotes the algebra of all bounded operators on a Hilbert space H. We show that every logmodular subalgebra of Mn(C) is unitary equivalent to an algebra of block upper triangular matrices, which was conjectured by V. Paulsen and M. Raghupathi. In particular, this shows that every unital contractive representation of a logmodular subalgebra of Mn(C) is automatically completely contractive. In the second part of the dissertation we investigate certain matrices composed of mixed, second?order moments of unitaries. The unitaries are taken from C??algebras with moments taken with respect to traces, or, alternatively, from matrix algebras with the usual trace. These sets are of interest in light of a theorem of E. Kirchberg about Connes' embedding problem and provide a new approach to it. Finally, we give a modification of I. Klep and M. Schweighofer?s algebraic reformulation of Connes' embedding problem by considering the ?-algebra of the countably generated free group. This allows us to consider only quadratic polynomials in unitary generators instead of arbitrary polynomials in self-adjoint generators.

We investigate the genericity of measure-preserving actions of the free group Fn, on possibly countably infinitely many generators, acting on a standard probability space. Specifically, we endow the space of all measure-preserving actions of Fn acting on a standard probability space with the weak topology and explore what properties may be verified on a comeager set in this topology. In this setting we show an analog of the classical Rokhlin Lemma. From this result we conclude that every action of Fn may be approximated by actions which factor through a finite group. Using this finite approximation we show the actions of Fn, which are rigid and hence fail to be mixing, are generic. Combined with a recent result of Kerr and Li, we obtain that a generic action of Fn is weak mixing but not mixing. We also show a generic action of Fn has sigma-entropy at most zero. With some additional work, we show the finite approximation result may be used to that show for any action of Fn, the crossed product embeds into the tracial ultraproduct of the hyperfinite II1 factor. We conclude by showing the finite approximation result may be transferred to a subspace of the space of all topological actions of Fn on the Cantor set. Within this class, we show the set of actions with sigma-entropy at most zero is generic.