Differential geometry on Thompson's components of positive operators

Consider the algebra L(H) of bounded linear operators on a Hilbert space H, and let L(H)(+) be the set of positive elements of L(H). For each A epsilo n L(H)(+) we study differential geometry of the Thompson component of A, C- A = {B epsilon L(H)(+) : A less than or equal to rB and B less than or equa l to sA for some s, r > 0}. The set of components is parametrized by means of all operator ranges of H. Each C-A is a differential manifold modelled i n an appropriate Banach space and a homogeneous space with a natural connec tion. Morover, given arbitrary B, C epsilon C-A, there exists a unique geod esic with endpoints B and C. Finally, we introduce a Finsler metric on C-A for which the geodesics are short and we show that it coincides with the so -called Thompson metric.