Grassmann manifolds of Jordan algebras (Q2501151)

The aim of this paper is to study manifolds of projections in JB-algebras using only real Jordan algebraic structures. The author shows that, in any JB-algebra, the projections form a real Banach manifold \({\mathcal P}\), and, in a JBW-algebra, the finite rank projections \({\mathcal P}_f\) and the infinite rank projections \({\mathcal P}_{\infty}\) are submanifolds of \({\mathcal P}\). In a JBW-factor, the manifold \({\mathcal P}_f\) consists of a sequence of connected components \(\{{\mathcal P}_n\}\), \(n \geq 0\), where \({\mathcal P}_n\) is the set of rank-\(n\) projections. Moreover, each of these components carries the structure of a Riemannian symmetric space. This result generalizes \textit{U.~Hirzebruch}'s result [Math.\ Z.\ 90, 339--354 (1965; Zbl 0139.39202)] on the manifold of minimal projections in a (finite-dimensional) formally real simple Jordan algebra.