Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces / Volkmar Liebscher.

Format:

Book

Author/Creator:

Liebscher, Volkmar, 1965- author.

Series:

Memoirs of the American Mathematical Society ; Volume 199, Number 930.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 199, Number 930

Language:

English

Subjects (All):

Hilbert space.

Random sets.

Invariants.

Calculus of tensors.

Physical Description:

1 online resource (124 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2009.

Language Note:

English

Summary:

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 E0-semigroups upto cocyle 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 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 In, IIn and 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.

Contents:

""Contents""; ""Chapter 1. Introduction""; ""Chapter 2. Basics""; ""Chapter 3. From Product Systems to Random Sets""; ""3.1. Product Systems""; ""3.2. Random Sets in Product Systems""; ""3.3. Measure Types as Invariants""; ""3.4. Measure Types Related to Units""; ""3.5. Tensor Products (I)""; ""Chapter 4. From Random Sets to Product Systems""; ""4.1. General Theory""; ""4.2. Example 1: Finite Random Sets""; ""4.3. Example 2: Countable Random Sets""; ""4.4. Example 3: Random Cantor Sets""; ""4.5. Tensor Products (II)""; ""4.6. The map e [omitted] M[sup(e,u)] is surjective""

""Chapter 5. An Hierarchy of Random Sets""""5.1. Factorising Projections and Product Subsystems""; ""5.2. Subsystems of e[sup(M)]""; ""5.3. The Lattice of Stationary Factorising Measure Types""; ""Chapter 6. Direct Integral Representations""; ""6.1. Random Sets and Direct Integrals""; ""6.2. Direct Integrals in Product Systems""; ""6.3. Characterisations of Type I Product Systems""; ""6.4. Unitalising Type III Product Systems""; ""Chapter 7. Measurability in Product Systems: An Algebraic Approach""; ""7.1. GNS-representations""

""7.2. Algebraic Product Systems and Intrinsic Measurable Structures""""7.3. Product Systems of W*-Algebras""; ""7.4. Product systems and Unitary Evolutions""; ""7.5. Additional Results on Measurability""; ""Chapter 8. Construction of Product Systems from General Measure Types""; ""8.1. General Results""; ""8.2. Product Systems from Random Sets""; ""8.3. Product Systems from Random Measures""; ""8.4. Product Systems from Random Increment Processes""; ""Chapter 9. Beyond Separability: Random Bisets""; ""Chapter 10. An Algebraic Invariant of Product Systems""

""Chapter 11. Conclusions and Outlook""""Bibliography""

Notes:

"Volume 199, Number 930 (first of 6 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0536-9