Whether N arising from a vNOE coupling is *-closed

Determine whether the subset N = {a ∈ A : a ⊗ 1 ∈ V} defined via a vNOE coupling is closed under the *-operation; equivalently, show whether N is a *-subalgebra of A, where V is the L^2(A ⊗ Q)-closure of the linear span of {φ(b) x : b ∈ B, x ∈ Q} for a normal *-homomorphism φ : B → A ⊗ Q satisfying E_Q ∘ φ = τ_B.

Background

In Lemma ‘special subalgebra of A via coupling’, the authors show that N = {a ∈ A : a ⊗ 1 ∈ V} is an SOT-closed subalgebra (closed under multiplication) of A, where V is generated by the vNOE coupling maps. However, they do not establish whether N is closed under taking adjoints.

Clarifying whether N is *-closed would sharpen the structural understanding of subalgebras arising from vNOE couplings and could influence how such couplings interact with *-algebraic properties.