Semifinite tracial subalgebras

Abstract

Let $\mathcal{M}$ be a semifinite von Neumann algebra, and let $\mathcal{A}$ be a tracial subalgebra of $\mathcal{M}$. We show that $\mathcal{A}$ is a subdiagonal algebra of $\mathcal{M}$ if and only if it has the unique normal state extension property and is a $\tau$-maximal tracial subalgebra, which is also equivalent to $\mathcal{A}$ having the unique normal state extension property and satisfying $L_2$-density.