Implications between the two defining properties of properness in the quantum setting

Determine whether, for a right action α of a locally compact quantum group G on a discrete quantum space (M, δ), the condition that u_{x,y} ∈ Cr(G) for all x, y ∈ M0 implies that u_{x,y} lies in the domain of the left Haar weight φ for all x, y ∈ M0, and whether the converse implication holds.

Background

The paper defines a right action α of a locally compact quantum group G on a discrete quantum space (M, δ) to be proper if, for all x, y in the finitely supported subalgebra M0, the coefficients u_{x,y} := (δ(x*·) ⊗ id)α(y) lie both in the reduced C*-algebra Cr(G) and in the domain of the left Haar weight. In the classical case (G a locally compact group acting on a discrete set), each of these two properties forces properness in the usual sense, and hence they are equivalent.

However, in the quantum setting the paper notes that it is not known whether one of these two properties implies the other. Resolving this would clarify whether a single condition could be used to characterise properness of actions on discrete quantum spaces for locally compact quantum groups.