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The Karcher mean on the cone Ω 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 L1-probability measures on Ω. 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.

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Journal of Mathematical Analysis and Applications