Poncaré half-space of a C*-algebra

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Let A be a C*-algebra. Given a representation A⊂ B(L) in a Hilbert space L, the set G+⊂ A of positive invertible elements can be thought as the set of inner products in L, related to A, which are equivalent to the original inner product. The set G+ has a rich geometry, it is a homogeneous space of the invertible group G of A, with an invariant Finsler metric. In the present paper we study the tangent bundle TG+ of G+, as a homogenous Finsler space of a natural group of invertible matrices in M₂(A), identifying TG+ with the Poincaré halfspace H of A, H={h∈ A: Im(h)≥ 0, Im(h) invertible}. We show that ≃ TG+ has properties similar to those of a space of non-positive constant curvature.