Equality of intrinsic and dynamical anticoarse spaces

Establish whether, for any inclusion N ≤ M with ϕ-expectation and faithful normal semifinite weight ϕ on M, the intrinsic ϕ-anticoarse space L†(N ≤ M, ϕ) (Definition 3.1) coincides with the dynamical ϕ-anticoarse space L†,dyn(N ≤ M, ϕ) (Definition 3.8).

Background

The paper develops two versions of anticoarse spaces in the non-tracial setting to analyze indecomposability phenomena: an intrinsic definition directly in terms of N–N bimodule maps on L^2(M,ϕ), and an extrinsic ("dynamical") definition using the continuous core and its L^2-structure.

Both notions are used for different technical reasons in subsequent results, but a unification would simplify the framework and strengthen conclusions. The authors express that these spaces are expected to coincide in many cases, but this is not currently known in general.