Analysis and Quantum Groups

by Lars Tuset

Overview

This volume presents a completely self-contained introduction to the elaborate theory of locally compact quantum groups, bringing the reader to the frontiers of present-day research. The exposition includes a substantial amount of material on functional analysis and operator algebras, subjects which in themselves have become increasingly important with the advent of quantum information theory. In particular, the rather unfamiliar modular theory of weights plays a crucial role in the theory, due to the presence of ‘Haar integrals’ on locally compact quantum groups, and is thus treated quite extensively

The topics covered are developed independently, and each can serve either as a separate course in its own right or as part of a broader course on locally compact quantum groups. The second part of the book covers crossed products of coactions, their relation to subfactors and other types of natural products such as cocycle bicrossed products, quantum doubles and doublecrossed products. Induced corepresentations, Galois objects and deformations of coactions by cocycles are also treated. Each section is followed by a generous supply of exercises. To complete the book, an appendix is provided on topology, measure theory and complex function theory.

Product Details

ISBN-13:	9783031072468
Publisher:	Springer-Verlag New York, LLC
Publication date:	07/27/2022
Sold by:	Barnes & Noble
Format:	eBook
File size:	39 MB
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About the Author

Lars Tuset is Professor of Mathematics at Oslo Metropolitan University. He has worked actively in the area of operator algebras and quantum groups since locally compact quantum groups were introduced in the 90’s, and offers firsthand experience of the development of the theory with all its pros and cons. He has previously published a monograph on compact quantum groups and their representation categories.

His research includes amenability of locally compact quantum groups, and cocycle deformations of coactions of locally compact quantum groups in the context of crossed products.

Table of Contents

Preface.- Set theoretic preliminaries.- Banach spaces.- Bases in Banach spaces.- Operators on Hilbert spaces.- Spectral theory.- States and  representations.- Types of von Neumann algebras.- Tensor products.- Unbounded operators.- Tomita-Takesaki theory.- Spectra and type III factors.- Quantum groups and  duality.- Special cases.- Classical crossed products.- Crossed  products for quantum groups.- Generalized and continuous crossed products.- Basic construction and quantum groups.- Galois objects and  cocycle deformations.- Doublecrossed  products of quantum groups.- Induction.-Appendix.- Bibliography.- Index.- Exercises.