Approximations in Operator Theory and Free Probability

We will investigate several related problems in Operator Theory and Free Probability. The notion of an exact C*-algebra is modified to reduced free products where it is shown, by examining another exact sequence of Toeplitz-Pimsner Algebras, that every C*-algebra is freely exact. This enables a discussion of strongly convergent random variables where we show that strong convergence is preserved under reduced free products. We will also analyze the distributions of freely independent random variables where it is shown that the distribution of a non-trivial polynomial in freely independent semicircular variables is atomless and has an algebraic Cauchy transform. These results are obtained by considering an analogue of the Strong Atiyah Conjecture for discrete groups and by considering algebraic formal power series in non-commuting variables respectively. More information about the distributions of operators will be obtained by examining when normal operators are limits of nilpotent operators in various C*-algebras including von Neumann algebras and unital, simple, purely infinite C*-algebras. The main techniques used to examine when a normal operator is a limit of nilpotent operators come from known matrix algebra results along with the projection structures of said algebras. Finally, using specific information about norm convergence of nilpotent operators, we will examine the closed unitary and similarity orbits of normal operators in von Neumann algebras and unital, simple, purely infinite C*-algebras