2018/12/29 by Gao, Mingchu
#46L54 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1812.11259
This paper is devoted to studying R-diagonal and η-diagonal pairs of random variables. We generalize circular elements to the bi-free setting, defining bi-circular element pairs of random variables, which provide examples of R-diagonal pairs of random variables. Formulae are given for calculating the distributions of the product pairs of two *-bi-free R-diagonal pairs. When focusing on pairs of left acting operators and right acting operators from finite von Neumann algebras in the standard form, we characterize R-diagonal pairs in terms of the *-moments of the random variables, and of distributional invariance of the random variables under multiplication by free unitaries. We define η-diagonal pairs of random variables, and give a characterization of η-diagonal pairs in terms of the *-distributions of the random variables. If every non-zero element in a *-probability space has a non-zero *-distribution, we prove that the unital algebra generated by a 2× 2 off-diagonal matrix with entries of a non-zero random variable x and its adjoint x^* in the algebra and the diagonal 2× 2 scalar matrices can never be Boolean independent fromm the 2× 2 scalar matrix algebra with amalgamation over the diagonal scalar matrix algebra.