| Sumario: | 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 ∈ L(H)+ we study differential geometry of the Thompson component of A, CA = {B ∈ 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 ∈ CA, there exists a unique geodesic with endpoints B and C. Finally, we introduce a Finsler metric on CA for which the geodesics are short and we show that it coincides with the so-called Thompson metric.
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