Some Results on the Distributions of Operator Valued Semicircular Random Variables

Abstract

The operator-valued free central limit theorem and operator-valued semi-circular random variables were first introduced by D. Voiculescu in 1995 as operator-valued free analogues of the classical central limit theorem and normal random variables, respectively. In 2007, R. Speicher and others showed that the operator-valued Cauchy transform of the semicircular distribution satisfies a functional equation involving the variance of the semicircular distribution. In this thesis, we consider a non - commutative probability space (A,EB,B) where in which A is a unital C*-algebra, B is a C*-subalgebra of A containing its unit and EB: A → B is a conditional expectation. For a given B−valued self-adjoint semicircular random variable S ∈ A with variance η, it is still an open question under what conditions the distribution of S has an atomic part. We provide a partial answer in terms of properties of η when B is the algebra of n × n complex matrices. In addition, we show that for a given compactly supported probability measure its associated Cauchy transform can be represented in terms of the operator-valued Cauchy transforms of a sequence of finite dimensional matrix-valued semicircular random variables in two ways. Finally, we give another representation of its Cauchy transform in terms of operator-valued Cauchy transform of an in finite dimensional matrix-valued semicircular random variable.