The Poincaré half-space of a C∗-algebra

Abstract

Let A be a unital C*-algebra. Given a faithful representation A⊂B(L) in a Hilbert space L, the set G^+⊂A of positive invertible elements can be thought of 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 M_2(A), identifying TG^+ with the it Poincaré half-space H of A. H={H ∈ A : Im(h)≥ 0,Im(h) invertible}. We show that H≃TG^+ has properties similar to those of a space of non-positive constant curvature.