Barycentric maps for compactly supported measures

After outlining the way in which an intrinsic mean G={G_n} on a complete metric space gives rise to a contractive barycentric map on some class of Borel probability measures and some basic examples of this process, we show how the resulting barycentric map gives rise to a general theory of integration of measurable functions into the space. We apply this machinery to the cone of positive invertible elements of a C^⁎-algebra equipped with the Thompson metric to derive barycentric maps and their basic properties arising from the power means. Finally we derive basic results for the Karcher barycenter including its approximation by the barycentric maps for power means and its satisfaction of the Karcher equation.