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.
Operator Theory, Functional Analysis, Mathematical Physics, Complex Analysis, Stochastic Analysis, Quantum Mechanics0