Analysis and quantum groups

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.

3.4 Dual Bases -- 3.5 The James Space J -- Exercises -- 4 Operators on Hilbert Spaces -- 4.1 Hilbert Spaces -- 4.2 Fourier Transform Over the Reals -- 4.3 Fourier Series -- 4.4 Polar Decomposition of Operators on Hilbert Spaces -- 4.5 Compact Normal Operators -- 4.6 Fredholm Operators -- 4.7 Traceclass and Hilbert-Schmidt Operators -- Exercises -- 5 Spectral Theory -- 5.1 Spectral Theory for Banach Algebras -- 5.2 Spectral Theory for C*-Algebras -- 5.3 Ideals and Hereditary Subalgebras -- 5.4 The Borel Spectral Theorem -- 5.5 Von Neumann Algebras -- 5.6 The -Weak Topology

Intro -- Preface -- Contents -- 1 Introduction -- 2 Banach Spaces -- 2.1 Normed Spaces -- 2.2 Operators on Banach Spaces -- 2.3 Linear Functionals -- 2.4 Weak Topologies -- 2.5 Extreme Points -- 2.6 Fixed Point Theorems -- 2.7 The Eberlein-Krein-Smulian Theorems -- 2.8 Reflexivity and Functionals Attaining Extreme Values -- 2.9 Compact Operators on Banach Spaces -- 2.10 Complemented and Invariant Subspaces -- 2.11 An Approximation Property -- 2.12 Weakly Compact Operators -- Exercises -- 3 Bases in Banach Spaces -- 3.1 Schauder Bases -- 3.2 Unconditional Convergence -- 3.3 Equivalent Bases

5.7 The Kaplansky Den…