Equivalence between proper actions on discrete quantum spaces and existence of an algebraic quantum group core subalgebra

Prove that for every locally compact quantum group G, the following are equivalent: (1) There exists a discrete quantum space (M, δ) and a proper δ-preserving right action α of G on M such that for all x, y in the finitely supported subalgebra M0 the coefficients u_{y,x} := (δ(y*·) ⊗ id)α(x) lie in the reduced C*-algebra Cr(G) and in the domain of the left Haar weight; and (2) There exists a strongly dense *-subalgebra O(G) ⊂ L∞(G) contained in the domains of the Haar weights for which the coproduct Δ restricts to O(G) and makes (O(G), Δ) an algebraic quantum group in the sense of Van Daele and Kustermans–Van Daele.

Background

The paper proves that if a locally compact quantum group G acts properly on a discrete quantum space (M, δ), then one can canonically construct a strongly dense *-subalgebra O(G) ⊂ L∞(G), lying in the domains of both Haar weights, such that Δ|_{O(G)} equips O(G) with the structure of an algebraic quantum group.

In the classical group case, Landstad and Van Daele showed that a locally compact group admits such an algebraic quantum group description if and only if it contains a compact open subgroup; moreover, a proper action on a discrete space arises from the translation action on cosets of such a subgroup. Motivated by this, the conjecture posits the full equivalence for all locally compact quantum groups. The paper also notes that the direction (2 ⇒ 1) holds for classical groups, and that both directions hold for compact and discrete quantum groups, while the remaining direction (2 ⇒ 1) is open in general.