Maximality and finiteness of type 1 subdiagonal algebras

Let $\mathfrak A$ be a type 1 subdiagonal algebra in a $σ$-finite von Neumann algebra $\mathcal M$ with respect to a faithful normal conditional expectation $Φ$. We give necessary and sufficient conditions for which $\mathfrak A$ is maximal among the $σ$-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.

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