Lipschitz Continuity of the Absolute Value and Riesz Projections in Symmetric Operator Spaces

Abstract 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 ,  τ ), the absolute value mapping is Lipschitz continuous on the associated symmetric operator space E ( M ,  τ ) 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 p