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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
Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces
eBook ISBN:  978-1-4704-0536-6
Product Code:  MEMO/199/930.E
List Price: $70.00
MAA Member Price: $63.00
AMS Member Price: $42.00
Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces
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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
eBook ISBN:  978-1-4704-0536-6
Product Code:  MEMO/199/930.E
List Price: $70.00
MAA Member Price: $63.00
AMS Member Price: $42.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1992009; 101 pp
    MSC: Primary 60; Secondary 81

    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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1992009; 101 pp
MSC: Primary 60; Secondary 81

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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.