Pálfia Miklós (Szöul): Existence and uniqueness of the L1-Karcher mean

We study the Karcher equation corresponding to probability measures on the Borel sigma algebra of positive operators on a Hilbert space with the Thompson metric.
We develop an ODE flow theory for the Karcher equation of L1 probability measures, in order to establish existence and uniqeness of its solution.
The ODE curves solving in the strong sense the Cauchy problem attached to the Karcher equation are exponentially contracting, hence establishing the uniqueness of stationary points.
We establish the existence of the stationary point by approximating an L1 probability measure by finitely supported measures. We investigate a Trotter-Kato type product formula in this setting, leading to a law of large numbers.