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

TL;DR

This paper links stable random fields with von Neumann algebras, showing that the group measure space construction is an invariant and characterizing ergodicity via the algebra's central decomposition, revealing $W^*$-rigidity.

Contribution

It establishes the invariance of the group measure space construction for stable random fields and characterizes ergodicity through von Neumann algebra properties, connecting probability and operator algebras.

Findings

Group measure space construction is an invariant for minimal representations.

Ergodicity of stationary SαS fields is characterized by the von Neumann algebra's central decomposition.

Ergodicity is shown to be a $W^*$-rigid property for these 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 $\alpha$-stable (S$\alpha$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$\alpha$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.