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.