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.
Rezensionen / Stimmen
"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)
"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)
Produkt-Info
Reihe
Auflage
Sprache
Verlagsort
Verlagsgruppe
Springer International Publishing
Zielgruppe
Illustrationen
Maße
Höhe: 235 mm
Breite: 155 mm
Dicke: 22 mm
Gewicht
ISBN-13
978-3-030-46368-7 (9783030463687)
DOI
10.1007/978-3-030-46366-3
Schweitzer Klassifikation
Konrad Schmuedgen 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 Positivstellensaetze, 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).
Konrad Schmuedgen 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 Positivstellensaetze, 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 <i>Unbounded Self-adjoint Operators on Hilbert Space</i> (2012) and <i>The Moment Problem</i> (2017).
General Notation.- 1 Prologue: The Algebraic Approach to Quantum Theories.- 2 ?-Algebras.- 3 <i>O</i>*-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 Positivstellensaetze.- 11 The Operator Relation <i>X</i><i>X</i>*=<i>F</i>(<i>X</i>*<i>X</i>).- 12 Induced ?-Representations.- 13 Well-behaved ?-Representations.- 14 Representations on Rigged Spaces and Hilbert <i>C</i>*-modules. A Unbounded Operators on Hilbert Space.- B <i>C</i>*-Algebras and Representations.- C Locally Convex Spaces and Separation of Convex Sets.- References.- Symbol Index.- Subject Index.
