On bi-free probability and free entropy.

Free probability is a non-commutative analogue of probability theory. Recently, Voiculescu has introduced bi-free probability, a theory which aims to study simultaneously "left" and "right" non-commutative random variables, such as those arising from the left and right regular representations of a countable group. We introduce combinatorial techniques to characterise bi-free independence, generalising results of Nica and Speicher from the free setting to the bi-free setting. In particular, we develop the lattice of bi-non-crossing partitions which is deeply tied to the action of bi-freely independent random variables on a free product space. We use these techniques to show that a conjecture of Mastnak and Nica holds, and bi-free independence is equivalent to the vanishing of mixed bi-free cumulants vanishing. Moreover, we extend the theory into the operator-valued setting, introducing operator-valued cumulants which correspond to bi-freeness with amalgamation in the same way. Finally, we investigate regularity problems in algebras of non-commuting random variables. Using operator theoretic techniques show that in an algebra generated by non-commutative random variables which admit a dual system, any self-adjoint element with spectral measure singular with respect to Lebesgue measure is a multiple of 1. We are also able to slightly improve on prior results in the literature and show that any non-constant self-adjoint polynomial evaluated at a set of non-commutative random variables which are free, algebraic, and have finite free entropy must produce a variable with finite free entropy