Extend diffuseness equivalence of central sequence algebras to general inclusions

Determine whether, for an inclusion of C*-algebras B ⊂ A with associated inclusion of II₁ factors N = B'' ⊂ M = A'', the diffuseness of the relative central sequence algebra A' ∩ B^ω in the C*-setting (with respect to the ultrapower trace τ^ω) is equivalent to the diffuseness of the relative central sequence algebra N' ∩ M_ω in the von Neumann setting. Equivalently, ascertain whether the Kirchberg–Rørdam equivalence A' ∩ A^ω diffuse ⇔ M' ∩ M_ω diffuse extends from the single-algebra case to arbitrary inclusions B ⊂ A.

Background

The paper uses central sequence algebras to formulate conditions ensuring selflessness of inclusions. In the single-algebra case, Kirchberg–Rørdam’s Theorem 3.3 implies an equivalence between diffuseness of the C*-central sequence algebra A' ∩ A^ω and diffuseness of the von Neumann central sequence algebra M' ∩ M_ω, where M = A''.

Within the paper’s main technical framework, Case II assumes diffuseness of A' ∩ (P A)^ω ⊂ A^ω to construct required freely independent elements. The authors note that while the single-algebra equivalence is known, it is unclear if a corresponding equivalence persists for arbitrary inclusions of C*-algebras, motivating the stated open question.