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This volume deals with Gleason's theorem and Gleason's measures and indicates the many ways in which they can be applied. The book comprises five chapters. Chapter 1 is devoted to elements of Hilbert space theory. Chapter 2 is devoted to quantum logic theory. Gleason's theorem is described and proved in Chapter 3, together with proofs for measures that can attain infinite values. In Chapter 4 the possibility of applying Gleason's theorem to the completeness criteria of inner product spaces is addressed. Chapter 5 discusses orthogonal measures and the unexpected possibility of describing states on Keller spaces, as well as other applications. Throughout the book, important facts and concepts are illustrated exercises. For mathematicians and physicists interested in the mathematical foundations of quantum mechanics, and those whose work involves noncommutative measure theory, orthomodular lattices. Hilbert space theory and probability theory.
| Preface | |
| Introduction | p. 1 |
| Hilbert Space Theory | p. 7 |
| Inner Product Spaces | p. 7 |
| Maximal Orthonormal Systems and Bases | p. 12 |
| Linear Functionals and their Applications | p. 20 |
| Linear Operators | p. 34 |
| Special Classes of Operators | p. 45 |
| Unbounded Operators | p. 55 |
| C* -algebras and W* -algebras | p. 63 |
| Theory of Quantum Logics | p. 71 |
| Elements of Quantum Logics | p. 71 |
| Joint Distributions of Observables | p. 89 |
| Commutator and Partial Compatibility | p. 97 |
| Joint Distributions for Non-compatible Observables | p. 105 |
| Joint Distributions in L(H) | p. 119 |
| Gleason's Theorem | p. 129 |
| Proof of Gleason's Theorem | p. 131 |
| Measures and von Neumann Algebras | p. 151 |
| Gleason's Theorem for Infinite Measures | p. 154 |
| Cardinal Numbers and Gleason's Theorem | p. 161 |
| Classification and Examples of Infinite Measures | p. 164 |
| Measures Generated by Bilinear Forms | p. 167 |
| Gleason's Theorem for Signed Measures | p. 178 |
| The Hahn and Jordan Decomposition | p. 182 |
| Convergences of Gleason Measures | p. 188 |
| Gleason States on Tensor Products | p. 192 |
| Gleason's Theorem and Completeness Criteria | p. 203 |
| Algebraic Completeness Criteria | p. 204 |
| Measure-Theoretic Completeness Criteria | p. 213 |
| Gleason's Theorem and Regular Charges | p. 228 |
| Systems of Charges and Completeness | p. 235 |
| Applications of Gleason's Theorem | p. 245 |
| Sum Logics | p. 245 |
| Orthogonal Vector-Valued Measures | p. 256 |
| Orthogonal Measures and Hilbert Spaces | p. 266 |
| Gleason's Theorem and Keller Spaces | p. 276 |
| Bibliography | p. 293 |
| Index of Symbols | p. 321 |
| Index | p. 325 |
| Table of Contents provided by Blackwell. All Rights Reserved. |
ISBN: 9780792319900
ISBN-10: 0792319907
Series: Mathematics and Its Applications
Audience:
Professional
Format:
Hardcover
Language:
English
Number Of Pages: 341
Published: 31st January 1993
Dimensions (cm): 23.5 x 15.5
x 2.0
Weight (kg): 0.664
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