# Existence and Uniqueness of the Karcher Mean on Unital $C^*$-algebras

## TL;DR

This paper proves the existence, uniqueness, and real analyticity of the Karcher mean on unital $C^*$-algebras, extending previous results from the cone of invertible positive elements in operator algebras.

## Contribution

It establishes the existence and uniqueness of the Karcher mean on any unital $C^*$-algebra and proves its real analyticity, generalizing prior work on operator algebras.

## Key findings

- Karcher mean satisfies the Karcher equation and is unique.
- Karcher mean and barycenter are real analytic functions.
- Existence and uniqueness are established for all unital $C^*$-algebras.

## Abstract

The Karcher mean on the cone $\Omega$ of invertible positive elements of the $C^*$-algebra $\mathcal{B}(E)$ 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 boththe 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.