The basic objects in this paper are the positive definite cones of C ⁎ -algebras.
From certain geometric type means defined on such a cone, we create specific order
relations. From those order relations, we define certain distance measures. Finally,
to those distance measures we relate certain natural geometric type means. We will
find that starting with the Kubo-Ando geometric mean, at the end we arrive at the
Fiedler-Pták geometric mean and, conversely, if we start with the Fiedler-Pták geometric
mean, then at the end we arrive at the Kubo-Ando geometric mean. Moreover, if we start
with the Log-Euclidean mean, at the end we get to the same Log-Euclidean mean. Along
the way, we collect several known results and present new findings concerning the
so-called near order. Finally, we point out that some of the results presented in
the paper can also be used to provide characterizations of commutative C ⁎ -algebras.
