The Carathéodory Pseudodistance and Positive Linear Operators

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.

On Preserving the Kobayashi Pseudodistance

If X is a complex space, the Kobayashi pseudo-distance dx is an intrinsic pseudometric on X defined as follows. If p and q are points of X, a chain α from p to q consists of intermediate points p0, …, pr with p0 = p and pr = q together with maps fi of the unit disc D = {z ∈ Cl‖z| < 1} into X and points ai and bi in D such that fi(ai) = Pi−1 and fi(bi) = pi for i = 1, …, r.