Invariant random subgroups and action versus representation maximality

Let G be a countably infinite group and (X,μ) a standard non-atomic probability space. We denote by A(G,X, μ) the space of measure preserving actions of G on (X,μ) with the weak topology. If a,b ∈ A(G,X, μ), we say that a is weakly contained in b, in symbols a b, if a is in the closure of the set of isomorphic copies of b (i.e., it is in the closure of the orbit of b under the action of the automorphism group of (X,μ) on A(G,X, μ); see [K]). We say that a ∈ A(G,X, μ) is action-maximal if for all b ∈ A(G,X, μ) we have b a. Such a exist by a result of Glasner-Thouvenot-Weiss, Hjorth, see [K, Theorem 10.7]). Now let H be a separable, infinite-dimensional Hilbert space and denote by Rep(G,H) the space of unitary representations of G on H with the weak topology (see [K, Appendix H]). For π, ρ ∈ Rep(G,H) we denote by π ρ the usual relation of weak containment of representations (see [BHV], [K, Appendix H]). We say that π ∈ Rep(G,H) is representation-maximal if for all ρ ∈ Rep(G,H) we have ρ π. It is easy to check that such π exist. For any action a ∈ A(G,X, μ), let κ be the associated representation on L(X,μ), called the Koopman representation, and by κ0 its restriction to the orthogonal of the constant functions (see [K, page 66]). Then we have