This study consists of two projects on bi-free probability. In the first project, a bi-free central limit distribution is investigated. We find the principal function of the completely non-normal operator l(vv1) + l(vv1)* + i(r(vv2) * + r(v2) * ) on a subspace of the full Fock space F(H) which arises from a bi-free central limit distribution. By the fact that the principal function of a pure hyponormal operator with trace class self-commutator is an extension of the Fredholm index of the operator, we find the essential spectrum of this operator. In the second part, we examine the reduced bi-free product C*-algebra generated by two pairs of commuting self-adjoint projections. In particular, we partially describe how to find the bi-free product states and the corresponding C*-algebra given by the GNS construction for a generic distribution of the projections. We prove some general results analogous to Voiculescu's partial R- and S-transforms by using combinatorial techniques on bi-free setting.
