# Journal of Operator Theory

Volume 77, Issue 1, Winter 2017 pp. 87-107.

Contractive barycentric map

**Authors**:
Jimmie D. Lawson (1) and Yongdo Lim (2)

**Author institution:**(1) Department of Mathematics, Louisiana State
University, Baton Rouge, LA 70803, U.S.A.

(2) Department of Mathematics,
Sungkyunkwan University, Suwon 440-746, Korea

**Summary: **We first develop in the context of complete metric spaces a one-to-one correspondence between the class of means
$G=\{G_n\}_{n\geqslant 2}$ that are symmetric, multiplicative, and contractive and the class of contractive (with respect to the
Wasserstein metric) barycentric maps on the space of $L^1$-prob-ability measures. We apply this equivalence
to the recently introduced and studied Karcher mean
on the open cone $\mathbb{P}$ of positive invertible operators on a Hilbert space equipped with the Thompson metric
to obtain a corresponding contractive barycentric map. In this context we derive a version of earlier results of Sturm and
Lim and Palfia about approximating the Karcher mean with the more constructive inductive mean. This leads to the conclusion
that the Karcher barycenter lies in the strong closure of the convex hull of the support of a probability measure. This fact is a
crucial ingredient in deriving a version of Jensen's inequality, with which we close.

**DOI: **http://dx.doi.org/10.7900/jot.2015dec24.2111

**Keywords: **positive operator, operator mean, barycentric map, Karcher mean,
geodesic metric space, Jensen's inequality

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