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This work discovers a new link between probability theory (of stable randomfields) and von Neumann algebras. It is established that the group measurespace construction corresponding to a minimal representation is an invariant ofa stationary symmetric $\alpha$-stable (S$\alpha$S) random field indexed by anycountable group $G$. When $G=\mathbb{Z}^d$, we characterize ergodicity (andalso absolute non-ergodicity) of stationary S$\alpha$S fields in terms of thecentral decomposition of this crossed product von Neumann algebra coming fromany (not necessarily minimal) Rosinski representation. This shows thatergodicity (or the complete absence of it) is a $W^\ast$-rigid property (in asuitable sense) for this class of fields. All our results have analogues forstationary max-stable random fields as well.

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