Existence and uniqueness of the Karcher mean on unital C*-algebras

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:483

Article

Lawson, Jimmie¹ 

[1] Louisiana State Univ, Dept Math, Baton Rouge, LA 70803 USA

关键词: Karcher mean and barycenter;    Karcher equation;    C*-algebra;    Integrable measures;    Bochner integral;    Real analytic;   

DOI  :  10.1016/j.jmaa.2019.123625

来源: Elsevier

【 摘 要 】

The Karcher mean on the cone Omega of invertible positive elements of the C*-algebra B(E) of bounded operators on a Hilbert space E has recently been extended to a contractive barycentric map on the space of L-1-probability measures on Omega. In this paper we first show that the barycenter satisfies the Karcher equation and then establish the uniqueness of the solution. Next we establish that the Karcher mean is real analytic in each of its coordinates, and use this fact to show that both the Karcher mean and Karcher barycenter map exist and are unique on any unital C*-algebra. The proof depends crucially on a recent result of the author giving a converse of the inverse function theorem. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free