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Projective spaces of a C*-algebra

E. Andruchow·G. Corach·D. Stojanoff
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Abstract

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.

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E. Andruchow, G. Corach, D. Stojanoff. 1999-11-18. Projective spaces of a C*-algebra. https://arxiv.org/abs/math/9911142

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