# Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

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*Volkmar Liebscher*

In a series of papers Tsirelson constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying \(E_0\)-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So the author connects each continuous tensor product system of Hilbert spaces with measure types of distributions of random (closed) sets in \([0,1]\) or \(\mathbb R_+\). These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the Tsirelson examples, that the classification scheme for product systems into types \(\mathrm{I}_n\), \(\mathrm{II}_n\) and \(\mathrm{III}\) is not complete. Moreover, based on a detailed study of this kind of measure types, the author constructs for each stationary factorising measure type a continuous tensor product system of Hilbert spaces such that this measure type arises as the before mentioned invariant.

#### Table of Contents

# Table of Contents

## Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

- Contents v6 free
- Chapter 1. Introduction ix10 free
- Chapter 2. Basics 116 free
- Chapter 3. From Product Systems to Random Sets 318
- Chapter 4. From Random Sets to Product Systems 1732
- Chapter 5. An Hierarchy of Random Sets 3146
- Chapter 6. Direct Integral Representations 4156
- Chapter 7. Measurability in Product Systems: An Algebraic Approach 5368
- Chapter 8. Construction of Product Systems from General Measure Types 7994
- Chapter 9. Beyond Separability: Random Bisets 89104
- Chapter 10. An Algebraic Invariant of Product Systems 93108
- Chapter 11. Conclusions and Outlook 97112
- Bibliography 99114