*Abstract: Recent advances in the axiomatic theory of operator means of probability measures and non-commutative - also called free - function theory lead us to investigate possible extensions of Loewner's theorem on operator monotone functions of a real variable going back to 1934. We provide characterizations of operator monotone and concave functions under minimal assumptions in several matrix or operator variables using LMIs and the theory of matrix convex sets. This generalizes results of Agler-McCarthy-Young providing characterizations restricted to commuting tuples of matrices, the case of $C^1$ real functions of several real variables. It also generalizes results of Pascoe-Tully-Doyle in the free setting. Our tools based on matrix convexity enable us to extend the domains of operator monotone functions, in particular to construct a free lift of a real operator monotone function of several real variables.*

*Host: John McCarthy*