# Random sets and invariants for (type II) continuous tensor product systems
of Hilbert spaces

### About this Title

**Volkmar Liebscher**

Publication: Memoirs of the American Mathematical Society

Publication Year
2009: Volume 199, Number 930

ISBNs: 978-0-8218-4318-5 (print); 978-1-4704-0536-6 (online)

DOI: http://dx.doi.org/10.1090/memo/0930

MathSciNet review: 2507929

MSC: Primary 46L55; Secondary 46L40, 46M05, 60G55

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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

**Chapters**

- Chapter 1. Introduction
- Chapter 2. Basics
- Chapter 3. From product systems to random sets
- Chapter 4. From random sets to product systems
- Chapter 5. An hierarchy of random sets
- Chapter 6. Direct integral representations
- Chapter 7. Measurability in product systems: An algebraic approach
- Chapter 8. Construction of product systems from general measure types
- Chapter 9. Beyond separability: Random bisets
- Chapter 10. An algebraic invariant of product systems
- Chapter 11. Conclusions and outlook