Wonhee Na Thesis Defense: Some results on bi-free probability

Date: May 25, 2018

Time: 10:30AM - 11:30AM

Location: BLOC 506A

Speaker: Wonhee Na

Description: This dissertation 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 $l(v_1)+l(v_1)^*+i(r(v_2)+r(v_2)^*)$ on a subspace of the full Fock space $\mathcal{F}(\mathcal{H})$ 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 describe the bi-free product states and the corresponding C*-algebra given by the GNS construction for all possible distributions 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.