On interrelations among different versions of a Heron type mean and commutativity in $C^*$-Algebras

The extension of the concept of a mean of positive real numbers to noncommutative settings, e.g., for Hilbert space operators, is a widely studied question. For example, in quantum information science, it is an important issue to find such extensions that fit the "best" to the studied physical problems. In fact, typically, there are many different ways of extension which, for commuting variables, all give the same value. In this paper, we are concerned with the converse: to what extent the coincidence of two extensions determines commutativity. Concretely, in our present work, we consider three different versions of the most common Heron type mean on the positive definite cone of a $C^*$-algebra: the Kubo-Ando type Heron mean, the naive or conventional version of the Heron mean, and the Wasserstein mean. We study equality relations among those objects and verify that they are closely connected to certain commutativity properties. They characterize either the commutativity of particular pairs of elements of a positive definite cone, or the centrality of positive definite elements, or the commutativity of the underlying algebra.