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 Lp-spaces, 1⩽p<∞, 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 Gâteaux differentiable in L2(M,τ) 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 Gâteaux differentiable in any Lp(M,τ), 1