eBook ISBN:  9781470405366 
Product Code:  MEMO/199/930.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 
eBook ISBN:  9781470405366 
Product Code:  MEMO/199/930.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 199; 2009; 101 ppMSC: 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


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
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