Maximality and finiteness of type 1 subdiagonal algebras

Abstract

Let A \mathfrak A be a type 1 subdiagonal algebra in a σ \sigma -finite von Neumann algebra M \mathcal M with respect to a faithful normal conditional expectation Φ \Phi . We give necessary and sufficient conditions for which A \mathfrak A is maximal among the σ \sigma -weakly closed subalgebras of M \mathcal M . In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of Arveson’s finiteness problem in 1967 in type 1 case.