Non-commutative Analysis

The book features new directions in analysis, with an emphasis on Hilbert space, mathematical physics, and stochastic processes. We interpret "non-commutative analysis" broadly to include representations of non-Abelian groups, and non-Abelian algebras; emphasis on Lie groups and operator algebras (C* algebras and von Neumann algebras.)

A second theme is commutative and non-commutative harmonic analysis, spectral theory, operator theory and their applications. The list of topics includes shift invariant spaces, group action in differential geometry, and frame theory (over-complete bases) and their applications to engineering (signal processing and multiplexing), projective multi-resolutions, and free probability algebras.

The book serves as an accessible introduction, offering a timeless presentation, attractive and accessible to students, both in mathematics and in neighboring fields.

Contents:

• Foreword

• Preface

• Acknowledgments

• Abbreviations, Notation, Some Core Theorems

• Introduction and Motivation:

  • Subjects and User's Guide

• Topics from Functional Analysis and Operators in Hilbert Space: A Selection:

  • Elementary Facts
  • Unbounded Operators in Hilbert Space
  • Spectral Theory

• Applications:

  • GNS and Representations
  • Completely Positive Maps
  • Brownian Motion
  • Lie Groups, and Their Unitary Representations
  • The Kadison-Singer Problem

• Extension of Operators:

  • Selfadjoint Extensions
  • Unbounded Graph-Laplacians
  • Reproducing Kernel Hilbert Space

• Appendices:

  • An Overview of Functional Analysis Books (Cast of Characters)
  • Terminology from Neighboring Areas
  • Often Cited
  • Prizes and Fame
  • List of Exercises
  • Definitions of Frequently Occurring Terms
  • List of Figures

• List of Tables

• Bibliography

• Index

Readership: Undergraduates, graduate students and researchers in operator theory, functional analysis, mathematical physics, complex analysis, stochastic analysis, quantum mechanics.