## Annals of Functional Analysis

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

Lawrence G. Brown

#### Abstract

In a previous article, we proved the equivalence of six conditions on a continuous function $f$ on an interval. These conditions determine 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 Akemann and Pedersen, 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 article 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. A treatment of other operator inequalities characterizing strong operator convexity is included.

#### Article information

Source
Ann. Funct. Anal., Volume 9, Number 1 (2018), 41-55.

Dates
Accepted: 4 February 2017
First available in Project Euclid: 29 June 2017

https://projecteuclid.org/euclid.afa/1498723217

Digital Object Identifier
doi:10.1215/20088752-2017-0023

Mathematical Reviews number (MathSciNet)
MR3758742

Zentralblatt MATH identifier
06841340

Subjects
Primary: 47A63: Operator inequalities
Secondary: 26A51: Convexity, generalizations

#### Citation

Brown, Lawrence G. A treatment of strongly operator-convex functions that does not require any knowledge of operator algebras. Ann. Funct. Anal. 9 (2018), no. 1, 41--55. doi:10.1215/20088752-2017-0023. https://projecteuclid.org/euclid.afa/1498723217

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