  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.
