This book is the first systematic treatment of measures on projection lattices of von Neumann algebras. It presents significant recent results in this field. One part is inspired by the Generalized Gleason Theorem on extending measures on the projection lattices of von Neumann algebras to linear functionals. Applications of this principle to various problems in quantum physics are considered (hidden variable problem, Wigner type theorems, decoherence functional, etc.). Another part of the monograph deals with a fascinating interplay of algebraic properties of the projection lattice with the continuity of measures (the analysis of Jauch-Piron states, independence conditions in quantum field theory, etc.). These results have no direct analogy in the standard measure and probability theory. On the theoretical physics side, they are instrumental in recovering technical assumptions of the axiomatics of quantum theories only by considering algebraic properties of finitely additive measures (states) on quantum propositions.
Table of ContentsPreface.
2: Operator Algebras. 2.1. C*-Algebras. 2.2. Von Neumann Algebras. 2.3. Jordan Algebras And Ordered Structures.
3: Gleason Theorem. 3.1. Reduction To Three-Dimensional Space. 3.2. Regularity Of Frame Functions On R3. 3.3. Boundedness Of Frame Functions. 3.4. Historical Remarks And Comments.
4: Completeness Criteria. 4.1. Functiona1 Completeness Criteria. 4.2. Algebraic Completeness Criteria. 4.3. Measure Theoretic Completeness Criteria. 4.4. Historical Remarks And Comments.
5: Generalized Gleason Theorem. 5.1. The Mackey-Gleason Problem. 5.2. Reduction To Scalar Quasi-Functionals. 5.3. Linear Extensions Of Measures On Type In Algebras. 5.4. Linear Extensions Of Measures On Infinite Algebras. 5.5. Linear Extensions Of Measures On Finite Algebras. 5.6. Historical Remarks And Comments.
6: Basic Principles Of Quantum Measure Theory. 6.1. Boundedness Of Completely Additive Measures. 6.2. Yosida-Hewitt Decompositions Of Quantum Measures. 6.3. Convergence Theorems. 6.4. Historical Remarks And Comments.
7: Applications Of Gleason Theorem. 7.1. Multiform Gleason Theorem And Decoherence. 7.2. Velocity Maps And Derivations. 7.3. Approximate Hidden Variables. 7.4. Historical Remarks And Comments.
8: Orthomorphisms Of Projections. 8.1. Orthomorphisms Of Projection Lattices. 8.2. Countable Additivity Of *-Homomorphisms. 8.3. Historical Remarks And Comments.
9: Restrictions And Extensions Of States. 9.1. Restriction Properties Of Pure States. 9.2. Gleason Type Theorems For Quantum Logics. 9.3. Historical Remarks And Comments.
10: Jauch-Piron States. 10.1. Basic Properties Of Jauch States. 10.2. Nonsingularity Of Jauch-Piron States. 10.3. Countable Additivity Of States. 10.4. Historical Remarks And Comments.
11: Independence Of Quantum Systems. 11.1. Independence In Classical And Quantum Theory. 11.2. Independence Of C*-Algebras. 11.3. Independence Of Von Neumann Algebras. 11.4. Historical Remarks And Comments.