# The royal road to automatic noncommutative real analyticity, monotonicity, and convexity

## TL;DR

This paper develops a framework to extend classical one-variable matrix analysis theorems on analyticity, monotonicity, and convexity to multiple noncommuting variables, simplifying proofs and broadening applicability.

## Contribution

It introduces the 'royal road theorem' that reduces multi-variable analyticity proofs to one-variable cases, and applies it to noncommutative L"owner and Kraus theorems.

## Key findings

- Established a general method for lifting one-variable analyticity results to multiple variables.
- Proved noncommutative L"owner and Kraus theorems over operator systems.
- Extended the 'butterfly realization' to general analytic functions in noncommutative settings.

## Abstract

It was shown classically that matrix monotone and matrix convex functions must be real analytic by L\"owner and Kraus respectively. Recently, various analogues have been found in several noncommuting variables. We develop a general framework for lifting automatic analyticity theorems in matrix analysis from one variable to several variables, the so-called "royal road theorem." That is, we establish the principle that the hard part of proving any automatic analyticity theorem lies in proving the one variable theorem. We use our main result to prove the noncommutative L\"owner and Kraus theorems over operator systems as examples, including an analogue of the "butterfly realization" of Helton-McCullough-Vinnikov for general analytic functions.