Image of the standard positive cone under the braided standard isomorphism

Characterize the image of the standard positive cone of the von Neumann algebra A1 ⊠ A2 inside L^2(A1) ⊗ L^2(A2) under the canonical unitary I_⊠ that identifies L^2(A1 ⊠ A2) with L^2(A1) ⊗ L^2(A2) for actions of a quasi-triangular locally compact quantum group. Provide an explicit description analogous to the known characterization in the ordinary tensor product case.

Background

The paper establishes a canonical unitary I_⊠ intertwining the standard representations and the unitary implementations for braided tensor products A1 ⊠ A2 arising from a quasi-triangular locally compact quantum group. While the unitary and modular conjugation are identified, the authors point out a gap: the structure of the standard positive cone under this identification is not determined.

In the ordinary tensor product case, the image of the positive cone in L^2(A1 ⊗ A2) is known (see Miura–Tomiyama, 1984). Extending such a characterization to the braided tensor product would complete the standard form picture developed in the paper.