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 homogeneous Finsler space of a natural group of invertible matrices in M2(A), identifying TG+ with the 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.