Noncommutative $H^2$ spaces

Let $\mathcal M$ be a von Neumann algebra with a faithful, finite, normal tracial state $\tau$, and let $\mathcal A$ be a finite, maximal subdiagonal algebra of $\mathcal M$. Let $H^2$ be the closure of $\mathcal A$ in the noncommutative Lebesgue space $L^2(\mathcal M ,\tau )$. Then $H^2$ possesses several of the properties of the classical Hardy space on the circle, including a commutant lifting theorem, some results on Toeplitz operators, an $H^1$ factorization theorem, Nehari's Theorem, and harmonic conjugates which are $L^2$ bounded.