Abstract
Let M \mathcal {M} be a von Neumann algebra with a faithful, finite, normal tracial state τ \tau , and let A \mathcal {A} be a finite, maximal subdiagonal algebra of M \mathcal {M} . Let H 2 H^2 be the closure of A \mathcal {A} in the noncommutative Lebesgue space L 2 ( M , τ ) L^2(\mathcal {M},\tau ) . Then H 2 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 H^1 factorization theorem, Nehari’s Theorem, and harmonic conjugates which are L 2 L^2 bounded.
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