On operator monotone and operator convex functions

We establish monotonicity and convexity criteria for a continuous function $f\colon\mathbb R^+\to\mathbb R$ with respect to any $C^*$-algebra. We obtain some estimates for noncompactness measure of $W^*$-algebra elements products differences. We also give a commutativity criterion for a positive $\tau$-measurable operator and a positive operator from a von Neumann algebra.