R-Diagonal and eta-Diagonal Pairs of Random Variables

Mingchu Gao ¸ßÃ÷èÆ  (Louisiana College)

10:00-11:00, Dec 11, 2018   Science Building A510

Abstract:

R-Diagonal and eta-Diagonal Pairs of Random Variables Mingchu Gao The class of R-diagonal elements has received quite a bit of attention in the free probability literature. In particular, R -diagonal elements were among the first examples of non-normal operators in R - probability spaces for which the Brown spectral measure was calculated explicitly by Haagerup and Larsen in 2000, and for which the Brown measure technique was used to find invariant subspaces by Sniady and Speicher in 2001. R-diagonal elements were characterized in terms of its *-moments, of distribution invariance under free Haar unitary multiplication, and of freeness with amalgamation over the diagonal matrix algebra by Nica, Shlyakhtenko, and Speicher in 2001. R-diagonal pairs of random variables were introduced in bi-free probability by Skoufranis in 2016, as an example to produce R-cyclic pairs of matrices of random variables, and $R$-cyclic pairs of matrices of random variables were characterized in terms of bi-freeness with amalgamation over the diagonal matrix algebra. In this talk, we shall present our recent work on R-diagonal pairs of random variables in the framework of standard forms of finite von Neumann algebras. We will characterize R-diagonal pairs of random variables in terms of distribution invariance under free Haar unitary multiplication, and of their *-moments, generating Nica, Shlyakhtenko, and Speicker's work in 2001 to bi-free probability. Given sufficient time, we will present our study on eta-diagonal pairs of random variables, generating Bercovici, Nica, Noyes, and Szpojankowski's work in 2018 to bi-Boolean probability, which was introduced by Gu and Skoufranis in 2017.