Date of Award

Spring 2004

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Liming Ge

Abstract

Certain (reduced) free product is introduced in the framework of operator spaces. Under the construction, the free product of preduals of von Neumann algebras agrees with the predual of the free product of von Neumann algebras. This answers a question asked by Effros affirmatively. An example is presented to show that the C*-algebra reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of the operator space reduced free product of the two C*-algebras.

Free Markov processes are also investigated in Voiculescu's free probability theory. This highly non-commutative notion generalizes that of free Brownian motion and free Levy processes. Some free Markov processes are realized as solutions to free stochastic differential equations driven by free Levy processes. A special and rather interesting kind of free Markov processes, free Ornstein-Uhlenbeck processes, is studied in some details. It is shown that a probability measure on R is free self-decomposable if and only if it is the stationary distribution of a stationary free Ornstein-Uhlenbeck process (driven by a free Levy process). Finally, the notion of free fractional Brownian motion is introduced. Examples of fractional free Brownian motion are given, which are based on creation and annihilation operators on full Fock spaces. It is proved that the Langevin equation driven by fractional free Brownian motion has a unique solution. We call the solution a fractional free Ornstein-Uhlenbeck process.

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