  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 fixed
projection p. The resulting space P(p) admits a rich geometrical structure as a
holomorphic manifold and a homogeneous reductive space of the invertible group
of A. Moreover, several metrics (chordal, spherical, pseudo-chordal,
non-Euclidean - in Schwarz-Zaks terminology) are considered, allowing a
comparison among P(p), the Grassmann manifold of A and the space of positive
elements which are unitary with respect to the bilinear form induced by the
reflection e = 2p-1. Among several metrical results, we prove that geodesics
are unique and of minimal length when measured with the spherical and
non-Euclidean metrics.

Operator Algebras