Abstract. Using techniques drawn from the classical theory of rearrangement invariant Banach function spaces we develop a duality theory in the sense of Köthe for symmetric Banach spaces of measurable operators affiliated with a semifinite von Neumann algebra equipped with a distinguished trace. A principal result of the paper is the identification of the Köthe dual of a given Banach space of measurable operators in terms of normality. IntroductionThe study of rearrangement invariant spaces of measurable functions has long been of central importance in many branches of real analysis as is clear from the monograph of Krein, Petunin, and Semenov [KPS] and the paper of Luxemburg [Lu]. Of equal importance in the study of spaces of compact operators in Hubert space is the notion of singular value [GK] which, for a given compact operator plays a similar role to that of the decreasing rearrangement of a measurable function. These ideas admit a common extension in the notion of a generalized decreasing rearrangement of an arbitrary selfadjoint operator affiliated with a given semifinite von Neumann algebra (with a distinguished trace) which has proved fruitful in many contexts [Gr, Ovl, 2, 3, FA1, 2, FK, Ye2, 3], among others. Based on the theory of noncommutative integration introduced by Segal [Se], Ovcinnikov [Ovl, 2] showed that the setting of semifinite von Neumann algebras with a trace provides a natural setting for various interpolation theorems, unifying in particular work of Calderón [Ca] in spaces of measurable functions with analogous results of Russu [Ru] for trace ideals. Of central importance in this work is the role played by the rearrangement invariant structure. Similar ideas occur in the subsequent work of Yeadon [Ye2, 3] motivated by the classical theory of Banach function spaces in the sense of Luxemburg and Zaanen [Lu, Zal]. More recently, a method of construction of rearrangement invariant Banach spaces of measurable operators has been given in [DDP1, 2] which is considerably more general than that permitted by the methods of [Ov2,Ye3] and it is the intention of this paper to develop these

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A principal result of the paper is that if E is a symmetric Banach function space on the positive half-line with the Fatou property then, for all semifinite von Neumann algebras (M, {), the absolute value mapping is Lipschitz continuous on the associated symmetric operator space E(M, {) with Lipschitz constant depending only on E if and only if E has non-trivial Boyd indices. It follows that if M is any von Neumann algebra, then the absolute value map is Lipschitz continuous on the corresponding Haagerup L p -space, provided 1

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