Journal of Operator Theory

Volume 40, Issue 2, Fall 1998 pp. 339-355.

Noncommutative $H^p$ spaces

Authors: Michael Marsalli (1), and Graeme West(2)
Author institution: (1) Mathematics Department, Campus Box 4520, Illinois State University, Normal, Illinois 61790--4520, U.S.A.
(2) Department of Mathematics, University of Witwatersrand, PO Wits 2050, South Africa

Summary: Let ${\cal M}$ be a von Neumann algebra equipped with a finite, normalised, normal faithful trace $\tau$ and let $H^\infty$ be a finite maximal subdiagonal subalgebra of ${\cal M}$. For $1 \leq p < \infty$ let $H^p$ be the closure of $H^\infty$ in the noncommutative Lebesgue space $L^p({\cal M})$. Then $H^p$ is shown to possess many of the properties of the classical Hardy space $H^p({\bbb T})$ of the circle, such as various factorisation results including a Riesz factorisation theorem, a Riesz-Bochner theorem on the existence and boundedness of harmonic conjugates, direct sum decompositions, and duality.

Keywords: von Neumann algebra, non-commutative Hardy space, subdiagonal algebra

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