Maximality and finiteness of type 1 subdiagonal algebras

Let $\mathfrak{A}$ be a type 1 subdiagonal algebra in a $\sigma$-finite von Neumann algebra $\mathcal{M}$ with respect to a faithful normal conditional expectation $\mathrm{\Phi }$. We give necessary and sufficient conditions for which $\mathfrak{A}$ is maximal among the $\sigma$-weakly closed subalgebras of $\mathcal{M}$. 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.