An Invitation to Unbounded Representations of ?-Algebras on Hilbert Space

by Konrad Schmüdgen

Overview

This textbook provides an introduction to representations of general ∗-algebras by unbounded operators on Hilbert space, a topic that naturally arises in quantum mechanics but has so far only been properly treated in advanced monographs aimed at researchers.

The book covers both the general theory of unbounded representation theory on Hilbert space as well as representations of important special classes of ∗-algebra, such as the Weyl algebra and enveloping algebras associated to unitary representations of Lie groups. A broad scope of topics are treated in book form for the first time, including group graded ∗-algebras, the transition probability of states, Archimedean quadratic modules, noncommutative Positivstellensätze, induced representations, well-behaved representations and representations on rigged modules.

Making advanced material accessible to graduate students, this book will appeal to students and researchers interested in advanced functional analysis and mathematical physics, and with many exercises it can be used for courses on the representation theory of Lie groups and its application to quantum physics. A rich selection of material and bibliographic notes also make it a valuable reference.

Product Details

ISBN-13:	9783030463656
Publisher:	Springer International Publishing
Publication date:	07/28/2020
Series:	Graduate Texts in Mathematics , #285
Edition description:	1st ed. 2020
Pages:	381
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Konrad Schmüdgen is Emeritus Professor at the Mathematical Institute of the University of Leipzig. He has worked for decades on unbounded representations and made important contributions. Among these are trace representation theorems for linear functionals, noncommutative Positivstellensätze, results on the transition probability, the theory of induced and well-behaved representations and classifications results of representations of special classes of algebras. He is the author of several books, including the Graduate Texts in Mathematics Unbounded Self-adjoint Operators on Hilbert Space (2012) and The Moment Problem (2017).

Table of Contents

General Notation.- 1 Prologue: The Algebraic Approach to Quantum Theories.- 2 ∗-Algebras.- 3 O*-Algebras.- 4 ∗-Representations.- 5 Positive Linear Functionals.- 6 Representations of Tensor Algebras.- 7 Integrable Representations of Commutative ∗-Algebras.- 8 The Weyl Algebra and the Canonical Commutation Relation.- 9 Integrable Representations of Enveloping Algebras.- 10 Archimedean Quadratic Modules and Positivstellensätze.- 11 The Operator Relation XX*=F(X*X).- 12 Induced ∗-Representations.- 13 Well-behaved ∗-Representations.- 14 Representations on Rigged Spaces and Hilbert C*-modules. A Unbounded Operators on Hilbert Space.- B C*-Algebras and Representations.- C Locally Convex Spaces and Separation of Convex Sets.- References.- Symbol Index.- Subject Index.

Editorial Reviews

This is a fantastic book. The material it covers is wonderful and exciting. The book is well-written, and in fact pleasant to read. It takes a familiar subject —- that is unfortunately not as popular among mathematicians as it should be —- and presents it from a very evocative perspective. Kudos to Schmüdgen.” (Michael Berg, MAA Reviews, July 22, 2023)

“It is very well written, the style is pleasant and attractive, and the information can be used by beginners and by specialists as well. All chapters are accompanied by exercises and pertinent historical comments. … all researchers interested in representation theory may regard this work not only as a reference book but also as a source of inspiration for further development.” (Florian-Horia Vasilescu, zbMATH 1458.47002, 2021)

From the Publisher