Projective spaces of a C*-algebra

TL;DR

This paper explores the geometric structure of projective spaces associated with C*-algebras, analyzing their manifold properties, metrics, and geodesic behavior, extending classical matrix space concepts to a non-commutative setting.

Contribution

It introduces a novel geometric framework for projective spaces of C*-algebras, including metric comparisons and geodesic properties, based on prior matrix space studies.

Findings

P(p) is a holomorphic manifold and a homogeneous reductive space.

Multiple metrics are considered, including spherical and non-Euclidean.

Geodesics are unique and minimal under certain metrics.

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.