DIFFERENTIAL GEOMETRY ON THOMPSONS 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 (belongs to) L(H)+ we study differential geometry of the Thompson component of A, CA = {B (belongs to) L(H)+ : A <,= rB and B <,= sA for some s, r > 0}. The set of components is parametrized by means of all operator ranges of H. Each CA is a differential manifold modelled in an appropriate Banach space and a homogeneous space with a natural connection. Morover, given arbitrary B,C (belongs to) CA, there exists a unique geodesic with endpoints B and C. Finally, we introduce a Finsler metric on CA for which the geodesies are short and we show that it coincides with the so-called Thompson metric.