An Invitation to Unbounded Representations of ∗-Algebras on Hilbert Space
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Beschreibung
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.
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.
Provides an accessible introduction to basic results and notions of unbounded representation theory Contains an extensive study of representations of the Weyl algebra and the commutation relation of quantum mechanics Treats many topics in unbounded representation theory in book form for the first time
Autor*in
Konrad Schmüdgen
Themen in »An Invitation to Unbounded Representations of ∗-Algebras on Hilbert Space«
GNS construction Trace functionals Unbounded representations Weyl algebra Infinitesimal representations Quadratic modules Positivstellensatz Rigged spaces Hilbert C*-modules
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“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)
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