# The noncommutative L\"owner theorem for matrix monotone functions over operator systems

## TL;DR

This paper extends the classical L"owner theorem to the noncommutative setting, showing that matrix monotone functions over operator systems analytically continue to noncommutative upper half planes, generalizing previous multivariable results.

## Contribution

It introduces a noncommutative L"owner theorem for matrix monotone functions over operator systems, broadening the scope of classical and multivariable generalizations.

## Key findings

- Real free functions over operator systems extend analytically to noncommutative upper half planes.
- The results unify classical, multivariable, and noncommutative cases of L"owner's theorem.
- The approach uses the relaxed Agler, McCarthy, and Young theorem on locally matrix monotone functions.

## Abstract

Given a function $f: (a,b) \rightarrow \mathbb{R},$ L\"owner's theorem states $f$ is monotone when extended to self-adjoint matrices via the functional calculus, if and only if $f$ extends to a self-map of the complex upper half plane. In recent years, several generalizations of L\"owner's theorem have been proven in several variables. We use the relaxed Agler, McCarthy and Young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case. Specifically, we show that a real free function defined over an operator system must analytically continue to a noncommutative upper half plane as map into another noncommutative upper half plane.