Trace-independence of vNOE for finite von Neumann algebras

Ascertain whether the definition of von Neumann orbit equivalence for tracial von Neumann algebras is independent of the choice of tracial states outside the finite-factor case; that is, given finite von Neumann algebras A and B equipped with faithful normal tracial states τ_A and τ_B used in the vNOE definition via an A ⊗ Q–B bimodule and a bi-tracial cyclic vector, determine whether (A, τ_A) ∼_vNOE (B, τ_B) implies (A, τ_A′) ∼_vNOE (B, τ_B′) for any other faithful normal tracial states τ_A′ and τ_B′.

Background

The authors define vNOE for tracial von Neumann algebras using a coupling that explicitly depends on chosen traces τ_A and τ_B. While symmetry and transitivity are established, the dependence on traces raises the question of whether the equivalence notion is intrinsic to the algebras or varies with the choice of traces.

They note that in the finite factor case, trace uniqueness mitigates this issue, but in the general finite (non-factor) setting, it is unclear if the equivalence is trace-independent.