A treatment of strongly operator convex functions that does not require any knowledge of operator algebras

TL;DR

This paper presents an accessible treatment of strongly operator convex functions, providing equivalent conditions that do not require operator algebra knowledge, and extends the understanding of operator inequalities and differential criteria.

Contribution

It offers a new, operator-algebra-free characterization of strongly operator convex functions, making the theory more accessible to a broader audience.

Findings

Equivalent conditions for strongly operator convex functions without operator algebra theory

Integral representation stronger than usual for operator convex functions

Differential criterion for strong operator convexity

Abstract

In [B1, Theorem 2.36] we proved the equivalence of six conditions on a continuous function f on an interval. These conditions define a subset of the set of operator convex functions, whose elements are called strongly operator convex. Two of the six conditions involve operator-algebraic semicontinuity theory, as given by C. Akemann and G. Pedersen in [AP], and the other four conditions do not involve operator algebras at all. Two of these conditions are operator inequalities, one is a global condition on f, and the fourth is an integral representation of f stronger than the usual integral representation for operator convex functions. The purpose of this paper is to make the equivalence of these four conditions accessible to people who do not know operator algebra theory as well as to operator algebraists who do not know the semicontinuity theory. We also provide a similar treatment of one theorem from [B1] concerning (usual) operator convex functions. And in two final sections we give a somewhat tentative treatment of some other operator inequalities for strongly operator convex functions, and we give a differential criterion for strong operator convexity.