GROUP MEASURE SPACE CONSTRUCTION, ERGODICITY AND W ∗-RIGIDITY FOR STABLE RANDOM FIELDS

The paper’s Theorem 5.4 states exactly that for a fixed stationary SαS random field indexed by a countable group, the nonsingular actions arising in any two minimal representations are conjugate and therefore their group measure space constructions are isomorphic as von Neumann algebras; i.e., the “minimal group measure space construction” is an invariant of the field (see the statement and proof of Theorem 5.4, which proceeds via Theorem 5.2 and the implication conjugacy ⇒ orbit equivalence ⇒ W*-equivalence) . The key inputs used in the paper are: (i) the uniqueness structure of minimal representations, codified in equations (4.2)–(4.4) , (ii) the Rosiński representation form (4.5) in the stationary case , and (iii) the extension of Rosiński’s 1995 conjugacy result to general countable groups (Theorem 5.2) . The candidate model’s solution reaches the same conclusion but by a slightly different route: it constructs an isometry between the two Lα spaces from equality of scale parameters, invokes a weighted-composition (Lamperti-type) form for surjective Lα-isometries to identify an underlying measurable conjugacy map, and then uses conjugacy to deduce a canonical crossed-product isomorphism. This is consistent with the paper’s logic and with the standard uniqueness theory for minimal spectral representations (as cited in Rosiński 1994/1995), but relies on one auxiliary step not explicitly stated in the paper: that the closed linear span of the spectral family {ft} in a minimal representation is all of Lα, which is stronger than the ratio-σ-algebra minimality used in the paper’s proof sketch. Aside from that technical strengthening, the two arguments agree on the core facts (minimal representation uniqueness and conjugacy of the actions) and deliver the same result. Hence, both are correct, with the paper’s proof using a brief “imitate Rosiński (1995)” route and the model providing a more operator-theoretic isometry argument.