Maximality of ultraweakly closed subdiagonal subalgebras

Determine whether every ultraweakly closed subdiagonal subalgebra H of a finite von Neumann algebra M (with respect to a faithful normal conditional expectation Φ: M → D onto a von Neumann subalgebra D) is necessarily maximal among subdiagonal subalgebras. Either prove the reverse implication (ultraweakly closed ⇒ maximal) in full generality or construct a counterexample.

Background

Arveson introduced subdiagonal subalgebras as a noncommutative generalization of Hardy spaces, defined via a conditional expectation Φ onto a diagonal D. A maximal subdiagonal subalgebra is automatically ultraweakly closed, but the converse is unresolved in general. Exel proved maximality under additional hypotheses (notably the presence of a faithful normal trace τ with τ∘Φ = τ), which cover the needs of this paper. The broader question—whether ultraweak closure alone implies maximality—remains unsettled.