Title

Following the trail of the operator geometric mean

Document Type

Article

Publication Date

1-1-2020

Abstract

This paper traces the development of the theory of the matrix geometric mean in the cone of positive definite matrices and the closely related operator geometric mean in the positive cone of a unital C∗-algebra. The story begins with the two-variable matrix geometric mean, moves to the n-variable matrix setting, then to the extension to the positive cone of the C∗-algebra of operators on a Hilbert space, and even to general unital C∗-algebras, and finally to the consideration of barycentric maps on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a substantial interplay with operator monotone functions, geometrical notions in metric spaces, particularly the notion of nonpositive curvature, some probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means.

Publication Source (Journal or Book title)

Trends in Mathematics

First Page

143

Last Page

153

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