Maximality and finiteness of type 1 subdiagonal algebras

Let A \mathfrak A be a type 1 subdiagonal algebra in alttext="sigma"> σ encoding="application/x-tex">\sigma -finite von Neumann alttext="script M"> class="MJX-tex-caligraphic" mathvariant="script">M encoding="application/x-tex">\mathcal M with respect to faithful normal conditional expectation alttext="normal Phi"> mathvariant="normal">Φ<!-- Φ encoding="application/x-tex">\Phi . We give necessary and sufficient conditions for which is maximal among the -weakly closed subalgebras of In addition, we show that finite automatically gives positive answer Arveson’s finiteness problem 1967 case.