Differentiation of operator functions in non-commutative Lp-spaces

The principal results in this paper are concerned with the description of differentiable operator functions in the non-commutative L-p-spaces, 1 less than or equal topless than or equal to infinity, associated with semifinite von Neumann algebras. For example, it is established that if f : R --> R is a Lipschitz function, then the operator function f is Gateaux differentiable in L-2 (M, tau) for any semifinite von Neumann algebra M if and only if it has a continuous derivative. Furthermore, if f : R --> R has a continuous derivative which is of bounded variation, then the operator function f is Gateaux differentiable in any L-p (M, tau), 1 <p< infinity. (C) 2003 Elsevier Inc. All rights reserved.