Quantum Measures and Spaces

This book has evolved from lectures and seminars given by the author at different academic institutions during the years 1983-1998. It can be divided into four parts. Noncommutative measure theory is the theme of the first part of the book. The relevant quantum structures are algebraically introduced. This is then used in the axiomatic, geometric model discussed in the second part of the book, where old and partly new groups and finite-dimensional R, C, H-spaces or spheres are studied for particle-series, a bag and the four basic interactions of physics. The third part investigates infinite dimensional spaces, particularly Archimedean and non-Archimedean orthomodular spaces, which generalize classical Hilbert spaces. The last part of the book contains short reviews on related topics which are useful to have at hand. Audience: This volume will be of interest to graduate students and researchers whose work involves mathematics of physics relativity and gravitation, order, lattices, and algebraic structures. Content: Front Matter....Pages i-xi Front Matter....Pages xiii-xiii Introduction....Pages 1-9 Orthomodular Measures....Pages 11-25 Gleason’s Theorem....Pages 27-38 Gleason’s Theorem in Applications....Pages 39-50 Inner Products....Pages 51-59 Front Matter....Pages 61-61 Manifolds, Matrices and Lie Structures....Pages 63-82 CC—Springs....Pages 83-92 Spins....Pages 93-108 Gravitons and 4D—Bags....Pages 109-130 Front Matter....Pages 131-131 Complete Structures....Pages 133-138 Dimension Theory....Pages 139-151 Measures on Complete Structures....Pages 153-158 Coordinatizations....Pages 159-169 The Kakutani—Mackey Theorem....Pages 171-178 Keller spaces....Pages 179-187 Orthomodular Spaces....Pages 189-197 Front Matter....Pages 199-199 About the 3—sphere....Pages 201-208 The Spectral Theorem....Pages 209-218 Topology....Pages 219-222 Strong and Weak Topologies....Pages 223-231 Back Matter....Pages 243-346 ....Pages 233-241 This book has evolved from lectures and seminars given by the author at different academic institutions during the years 1983-1998. It can be divided into four parts. Noncommutative measure theory is the theme of the first part of the book. The relevant quantum structures are algebraically introduced. This is then used in the axiomatic, geometric model discussed in the second part of the book, where old and partly new groups and finite-dimensional R, C, H-spaces or spheres are studied for particle-series, a bag and the four basic interactions of physics. The third part investigates infinite dimensional spaces, particularly Archimedean and non-Archimedean orthomodular spaces, which generalize classical Hilbert spaces. The last part of the book contains short reviews on related topics which are useful to have at hand. __Audience:__