## Some Results on the Distributions of Operator Valued Semicircular Random Variables

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##### Date

2011-09-13##### Author

Soltanifar, Mohsen

##### Type

Thesis##### Degree Level

Masters##### 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;E_B;B) where in which A is a unital C -algebra, B is a C -subalgebra of A containing its unit and E_B A B is
a conditional expectation. For a given B−valued self-adjoint semicircular random variable s > A with variance ; 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 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.

##### Degree

Master of Science (M.Sc.)##### Department

Mathematics and Statistics##### Program

Mathematics##### Supervisor

Belinschi, Serban; Samei, Ebrahim##### Committee

Steel, Tom; Choi, Yemon; Soteros, Chris##### Copyright Date

August 2011##### Subject

Semicircular distributions, Atoms, Cauchy transform, Continued Fraction