Operators and representation theory: canonical models for algebras of operators arising in quantum mechanics

Palle E.T. Jorgensen (Eds.)0444703217, 9780444703217, 9780080872582

Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas. This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly elegant manner by the use of Lie algebras, extensions, and projective representations. In several cases, this Lie algebraic approach to questions in mathematical physics and C * -algebra theory is new; for example, the Lie algebraic treatment of the spectral theory of curved magnetic field Hamiltonians, the treatment of irrational rotation type algebras, and the Virasoro algebra. Also examined are C * -algebraic methods used (in non-traditional ways) in the study of representations of infinite-dimensional Lie algebras and their extensions, and the methods developed by A. Connes and M.A.

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Table of contents :
Content:
Edited by
Pages ii-iii

Copyright page
Page iv

Preface
Page v

Acknowledgements
Page vi

Chapter 1. Introduction and Overview
Pages 1-2

Chapter 2. Definitions and Terminology
Pages 3-10

Chapter 3. Operators in Hilbert Space
Pages 11-20

Chapter 4. The Imprimitivity Theorem
Pages 21-36

Chapter 5. Domains of Representations
Pages 37-69

Chapter 6. Operators in the Enveloping Algebra
Pages 71-122

Chapter 7. Spectral Theory
Pages 123-163

Chapter 8. Infinite-Dimensional Lie Algebras
Pages 165-269

Appendix: Integrability of Lie Algebras
Pages 271-284

References
Pages 285-329

Index
Pages 331-337