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Group measure space construction, ergodicity and $W^\ast$-rigidity for stable random fields

Parthanil Roy

Abstract

This work discovers a novel link between probability theory (of stable random fields) and von Neumann algebras. It is established that the group measure space construction corresponding to a minimal representation is an invariant of a stationary symmetric $α$-stable (S$α$S) random field indexed by any countable group $G$. When $G=\mathbb{Z}^d$, we characterize ergodicity (and also absolute non-ergodicity) of stationary S$α$S fields in terms of the central decomposition of this crossed product von Neumann algebra coming from any (not necessarily minimal) Rosinski representation. This shows that ergodicity (or the complete absence of it) is a $W^\ast$-rigid property (in a suitable sense) for this class of fields. All our results have analogues for stationary max-stable random fields as well.

Group measure space construction, ergodicity and $W^\ast$-rigidity for stable random fields

Abstract

This work discovers a novel link between probability theory (of stable random fields) and von Neumann algebras. It is established that the group measure space construction corresponding to a minimal representation is an invariant of a stationary symmetric -stable (SS) random field indexed by any countable group . When , we characterize ergodicity (and also absolute non-ergodicity) of stationary SS fields in terms of the central decomposition of this crossed product von Neumann algebra coming from any (not necessarily minimal) Rosinski representation. This shows that ergodicity (or the complete absence of it) is a -rigid property (in a suitable sense) for this class of fields. All our results have analogues for stationary max-stable random fields as well.

Paper Structure

This paper contains 20 sections, 20 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ergodic Theory of Nonsingular Actions
  3. Ergodic and Neveu Decompositions
  4. Conjugacy and Orbit Equivalence
  5. von Neumann Algebras
  6. Factors and Central Decomposition
  7. Group Measure Space Construction
  8. $W^\ast$-rigidity
  9. Stationary S$\alpha$S Random Fields
  10. Integral Representation
  11. Minimal Representation
  12. The Stationary Case: Rosiński Representation
  13. Linking Stable Fields with von Neumann Algebras
  14. Minimal Representations, Conjugacy and the Crossed Product Invariant
  15. Weak Mixing, Ergodicity and Complete Non-ergodicity
  16. ...and 5 more sections

Key Result

Theorem 1.1

The group actions arising in all minimal representations of a fixed stationary S$\alpha$S random field are $W^\ast$-equivalent, i.e, their group measure space constructions are isomorphic as von Neumann algebras.

Figures (1)

  • Figure 1: The relation between $W^\ast$-rigidities

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2: Existence of Ergodic Decomposition
  • Definition 2.3
  • Definition 2.4
  • Theorem 3.1: von Neumann's Bicommutant Theorem
  • Definition 3.2
  • ...and 49 more