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Gleason's Theorem and Its Applications : Mathematics and its Applications - Anatolij Dvurecenskij

Gleason's Theorem and Its Applications

Mathematics and its Applications


Published: 31st January 1993
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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.

Introductionp. 1
Hilbert Space Theoryp. 7
Inner Product Spacesp. 7
Maximal Orthonormal Systems and Basesp. 12
Linear Functionals and their Applicationsp. 20
Linear Operatorsp. 34
Special Classes of Operatorsp. 45
Unbounded Operatorsp. 55
C* -algebras and W* -algebrasp. 63
Theory of Quantum Logicsp. 71
Elements of Quantum Logicsp. 71
Joint Distributions of Observablesp. 89
Commutator and Partial Compatibilityp. 97
Joint Distributions for Non-compatible Observablesp. 105
Joint Distributions in L(H)p. 119
Gleason's Theoremp. 129
Proof of Gleason's Theoremp. 131
Measures and von Neumann Algebrasp. 151
Gleason's Theorem for Infinite Measuresp. 154
Cardinal Numbers and Gleason's Theoremp. 161
Classification and Examples of Infinite Measuresp. 164
Measures Generated by Bilinear Formsp. 167
Gleason's Theorem for Signed Measuresp. 178
The Hahn and Jordan Decompositionp. 182
Convergences of Gleason Measuresp. 188
Gleason States on Tensor Productsp. 192
Gleason's Theorem and Completeness Criteriap. 203
Algebraic Completeness Criteriap. 204
Measure-Theoretic Completeness Criteriap. 213
Gleason's Theorem and Regular Chargesp. 228
Systems of Charges and Completenessp. 235
Applications of Gleason's Theoremp. 245
Sum Logicsp. 245
Orthogonal Vector-Valued Measuresp. 256
Orthogonal Measures and Hilbert Spacesp. 266
Gleason's Theorem and Keller Spacesp. 276
Bibliographyp. 293
Index of Symbolsp. 321
Indexp. 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: 325
Published: 31st January 1993
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 24.13 x 15.88  x 2.54
Weight (kg): 0.66