Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces

A principal result of the paper is that ifEis a symmetric Banach function space on the positive half-line with the Fatou property then, for all semifinite von Neumann algebras (M, τ), the absolute value mapping is Lipschitz continuous on the associated symmetric operator spaceE(M, τ) with Lipschitz constant depending only onEif and only ifEhas 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 HaagerupL^p-space, provided 1<p<∞.