Projective spaces of a C*-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C*-algebra A with a f ixed projection p. The resulting space P(p) admits a rich geometrical struc ture as a holomorphic manifold and a homogeneous reductive space of the inv ertible group of A. Moreover, several metrics (chordal, spherical, pseudo-c hordal, nonEuclidean - in Schwarz-Zaks terminology) are considered, allowin g a comparison among P(p), the Grassmann manifold of A and the space of pos itive elements which are unitary with respect to the bilinear form induced by the reflection epsilon = 2p - 1. Among several metrical results, we prov e that geodesics are unique and of minimal length when measured with the sp herical and non-Euclidean metrics.