Abstract. Let M be a von Neumann algebra with a faithful, finite, normal tracial state τ , and let A be a finite, maximal subdiagonal algebra of M. Let H 2 be the closure of A in the noncommutative Lebesgue space L 2 (M, τ). 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.

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Let M be a von Neumann algebra with a faithful normal tracial state τ and let H ∞ be a finite maximal subdiagonal subalgebra of M. In previous work we defined a harmonic conjugate relative to H ∞ . Let H 1 be the closure of H ∞ in the noncommutative Lebesgue space L 1 (M). By analysing the behaviour of the harmonic conjugate in L 1 (M), we identify the dual space of H 1 as a concrete space of operators.

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We investigate the validity of the Boltzmann equation to predict the reflection and transmission coefficients for an intensity modulated laser beam passing through a microscopic medium consisting of discrete scatterers. For a one-dimensional model system we demonstrate that the Boltzmann equation works remarkably well for small modulation frequencies, even to describe a medium comprised of only 10 scatterers. Discrepancies can be found only if the modulation wavelength of the laser intensity is commensurate with the spacing between the scatterers and if the medium is sufficiently ordered.