This book reveals how univalent functions appear in quantum probability theory. Building upon the recently established one-to-one correspondence between Loewner theory and the theory of non-commutative additive processes, the author invites readers to explore the interplay between complex analysis, classical probability theory, and quantum probability theory. Monotone independence and its relations to classical, free, and Boolean independence underpin the development of ideas.
Beginning with essential concepts from classical probability theory and complex analysis, the book goes on to define a quantum probability space and introduce five notions of independence. From this foundation, the central chapters explore convolutions and their respective central limit theorems; univalent functions; classical Loewner chains on the unit disk; slit mappings; and the relationship between free hemigroups, Loewner chains, and nonlinear resolvents. The final chapter offers an outlook on higher dimensional generalizations, including several open problems. Exercises with solutions invite readers to engage with the material throughout.
Univalent Functions in Quantum Probability Theory is an essential resource at the intersection of previously distinct fields. Intended for graduate students and researchers alike, it assumes a solid foundation in real and complex analysis, with basic knowledge of classical probability theory and Hilbert spaces.
By:
Sebastian Schlessinger
Imprint: American Mathematical Society
Country of Publication: United States
Dimensions:
Height: 254mm,
Width: 178mm,
ISBN: 9781470481803
ISBN 10: 1470481804
Series: Mathematical Surveys and Monographs
Pages: 268
Publication Date: 13 December 2025
Audience:
College/higher education
,
Further / Higher Education
Format: Paperback
Publisher's Status: Active
Introduction Classical probability theory The complex toolbox Quantum probability theory Convolutions and additive processes Univalent functions Radial Loewner chains revisited Slit mappings Free hemigroups, Loewner chains, and nonlinear resolvents Graph products as quantum random walks Outlook on higher dimensional generalizations Compactness of the class $S$ Continuous extension of univalent functions Bibliography Index
Sebastian Schlessinger, University of Wuerzburg, Germany.
