The noncommutative Löwner theorem for matrix monotone functions over operator systems

Abstract

Given a function f : (a, b) → R, Löwner'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 Löwner's theorem have been proven in several variables. We use the relaxed Agler, M c Carthy, 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.

Monotonicity of the principal pivot transform

“…Part (2) comes integrating a formula for the derivative of the principal pivot transform from Proposition 2.2. In the case where each of the blocks are square matrices, condition (1) implies condition (2) in Theorem 1.1 by the noncommutative Löwner theorem [8,7,5,6]. We note that as the principal pivot transform is an automorphism of the block 2 by 2 matrix "upper half plane", under conjugation by a suitable Cayley transform it is conjugate to an automorphism of block 2 by 2 matrices which was studied in [3].…”

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

“…Noncommutative analogues of Löwner's theorem have previously been established in [37,33]. The culmination of this work appears in [38], where the following theorem was proved in perhaps the highest level of generality that one should expect (although that proof relies on the commuting theorem in [2] and is thus "unnatural").…”

“…In commuting variables, see [2,35]. In noncommuting variables, see [33,37], culminating in essentially the most general framework in [38], which we reprove here using the "royal road" as a shortcut. Convexity theorems are somewhat less generally developed [17,18,21,23,22,19,33].…”

Analytic continuation of concrete realizations and the McCarthy Champagne conjecture

“…We introduce a new algebraic version of the usual noncommutative Herglotz realization (as in [55]), the so-called Herglotz-Nouveau formula. Finally, we refer to the noncommutative Nevanlinna realization [48,46,49].…”

“…Then is positive semidefinite, (6.2) extends further to all pairs of positive matrices as inputs for w 1 , w 2 . As the cone of pairs of positive matrices is a free, convex set, the noncommutative Löwner theorem, see [50,Theorem 1.7] as well as [46,52,49], implies that F is globally matrix monotone on (0, ∞) 2 . In particular, Effros and Hansen prove that convex non-commutative perspectives arise from convex commutative perspectives.…”