The geometry of spaces of projections in \(C^*\)-algebras
Let \(A\) be a \(C^*\)-algebra with identity represented as an algebra of operators in a Hilbert space \(E\). The authors give the geometric structure of the space \(\mathbb{Q}\) of idempotent elements of \(A\). Particular attention is devoted to the way \(Q\) is fibred over the space \(P\) of all selfadjoint elements of \(Q\). The set \(Q\) inherits from \(A\) a differentiable structure with a natural connection. The resulting retraction \(\pi : Q \to P\) is given by the polar decomposition of the associated symmetries, and alternative characterization of the fibres \(\mathbb{Q}_ p = \pi^{-1} (p)\) are presented. The authors also considered metric properties. A Finsler structure naturally related to \(Q@> \pi>> P\) is introduced in \(Q\) and the final part is devoted to the study of geodesics in \(Q\).
geometric structure of the space of idempotent elements
scientific article; zbMATH DE number 495496