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Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces
Volkmar Liebscher, GSF-National Research Centre for Environment and Health, Neuherberg, Germany
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Memoirs of the American Mathematical Society
2009; 101 pp; softcover
Volume: 199
ISBN-10: 0-8218-4318-4
ISBN-13: 978-0-8218-4318-5
List Price: US$66
Individual Members: US$39.60
Institutional Members: US$52.80
Order Code: MEMO/199/930
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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

  • Introduction
  • Basics
  • From product systems to random sets
  • From random sets to product systems
  • An hierarchy of random sets
  • Direct integral representations
  • Measurability in product systems: An algebraic approach
  • Construction of product systems from general measure types
  • Beyond separability: Random bisets
  • An algebraic invariant of product systems
  • Conclusions and outlook
  • Bibliography
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