On Certain Order Properties of Non Kubo-Ando Means in Operator Algebras

In this paper we consider some important non Kubo-Ando means on positive definite cones of C*-algebras and investigate their relations to the usual (Lowner) order. We study two basic questions. First, when, on the positive definite cone of a C*-algebra, does such a mean sigma satisfy the inequality A <= A sigma B B <= B for any pair A, B of elements with A <= B. This requirement seems to be the most basic one a mean should fulfill even for numbers. Actually, we study stronger forms of that question and obtain results characterizing central elements and commutativity of C*-algebras. Second, we investigate monotonicity properties of those means with respect to the usual order which also turn to be closely related to the commutativity of the underlying algebras.

Automated summary

In this paper, we investigate important non Kubo-Ando means on the positive definite cones of C*-algebras and their relations to the usual (Lowner) order. We focus on whether these means satisfy the basic requirement and their monotonicity properties, and our results involve the central elements and commutativity of C*-algebras.