Combinatorial theory of the free product with amalgamation and operator-valued free probability theory
About this Title
Publication: Memoirs of the American Mathematical Society
Publication Year: 1998; Volume 132, Number 627
ISBNs: 978-0-8218-0693-7 (print); 978-1-4704-0216-7 (online)
MathSciNet review: 1407898
MSC: Primary 46L50; Secondary 81S25
Free probability theory, introduced by Voiculescu, has developed very actively in the last few years and has had an increasing impact on quite different fields in mathematics and physics. Whereas the subject arose out of the field of von Neumann algebras, presented here is a quite different view of Voiculescu's amalgamated free product. This combinatorial description not only allows re-proving of most of Voiculescu's results in a concise and elegant way, but also opens the way for many new results.
Unlike other approaches, this book emphasizes the combinatorial structure of the concept of “freeness”. This gives an elegant and easily accessible description of freeness and leads to new results in unexpected directions. Specifically, a mathematical framework for otherwise quite ad hoc approximations in physics emerges.
Graduate students, research mathematicians and physicists working in functional analysis.
Table of Contents
- I. Preliminaries on non-crossing partitions
- II. Operator-valued multiplicative functions on the lattice of non-crossing partitions
- III. Amalgamated free products
- IV. Operator-valued free probability theory
- V. Operator-valued stochastic processes and stochastic differential equations