On some algebraic properties related to Heron type operator means on positive definite cones of C⁎-algebras

Abstract

In this paper we consider certain algebraic properties concerning variants of the Heron mean on positive definite cones of general C⁎-algebras. Those variants are the Kubo-Ando type Heron mean and the Wasserstein mean. The main part of the investigation concerns associativity properties. We present a number of results that show how far operations related to those two kinds of means are from being associative. Many of our results can also be viewed as characterizations of central positive definite elements or as characterizations of commutative C⁎-algebras.