On Operator Monotone and Operator Convex Functions

We establish monotonicity and convexity criteria for a continuous function f : R+ -> R with respect to any C*-algebra. We obtain an estimate for the measure of noncompactness of the difference of products of the elements of a W*-algebra. We also give a commutativity criterion for a positive tau-measurable operator and a positive operator from a von Neumann algebra.