The Carathéodory Pseudodistance and Positive Linear Operators

Abstract

We give a new elementary proof of Lempert's theorem, which states that for convex domains the Carathéodory pseudodistance coincides with the Lempert function and thus with the Kobayashi pseudodistance. Moreover, we prove the product property of the Carathéodory pseudodistance. Our methods are functional analytic and work also in the more general setting of uniform algebras.