Chapter 1. Introduction ........................................ ix
Chapter 2. Basics ............................................... 1
Chapter 3. From Product Systems to Random Sets .................. 3
3.1. Product Systems ......................................... 3
3.2. Random Sets in Product Systems .......................... 5
3.3. Measure Types as Invariants ............................ 11
3.4. Measure Types Related to Units ......................... 13
3.5. Tensor Products (I) .................................... 15
Chapter 4. From Random Sets to Product Systems ................. 17
4.1. General Theory ......................................... 17
4.2. Example 1: Finite Random Sets .......................... 20
4.3. Example 2: Countable Random Sets ....................... 21
4.4. Example 3: Random Cantor Sets .......................... 22
4.5. Tensor Products (II) ................................... 24
4.6. The map  is surjective ....................... 27
Chapter 5. An Hierarchy of Random Sets ......................... 31
5.1. Factorising Projections and Product Subsystems ......... 31
5.2. Subsystems of  ..................................... 33
5.3. The Lattice of Stationary Factorising Measure Types .... 38
Chapter 6. Direct Integral Representations ..................... 41
6.1. Random Sets and Direct Integrals ....................... 41
6.2. Direct Integrals in Product Systems .................... 44
6.3. Characterisations of Type I Product Systems ............ 46
6.4. Unitalising Type III Product Systems ................... 50
Chapter 7. Measurability in Product Systems: An Algebraic
Approach ............................................ 53
7.1. GNS-representations .................................... 53
7.2. Algebraic Product Systems and Intrinsic Measurable
Structures ............................................. 55
7.3. Product Systems of W*-Algebras ......................... 66
7.4. Product systems and Unitary Evolutions ................. 70
7.5. Additional Results on Measurability .................... 76
Chapter 8. Construction of Product Systems from General
Measure Types ....................................... 79
8.1. General Results ........................................ 79
8.2. Product Systems from Random Sets ....................... 84
8.3. Product Systems from Random Measures ................... 85
8.4. Product Systems from Random Increment Processes ........ 86
Chapter 9. Beyond Separability: Random Bisets .................. 89
Chapter 10.An Algebraic Invariant of Product Systems ........... 93
Chapter 11.Conclusions and Outlook ............................. 97
Bibliography ................................................... 99
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