Manifolds of tripotents in \(JB^*\)-triples (Q1568811)

An element \(x\) of a \(JB^*\)-triple is called tripotent if \(\{xxx\}=x\). In this paper the authors describe basic geometric structures of manifolds of tripotents in \(JB^*\)-triples. Connections on manifolds of finite-rank tripotents are defined, and their geodesics described. Projections in a \(JB^*\)-algebra are tripotents. If \(p\) is such a projection and \(P(p)\) and \(T(p)\) denote the connected components of \(p\) in the set of projections and tripotents, respectively, it is proved that \(P(p)\) is a real analytic direct submanifold of \(T(p)\), and the relationship between the corresponding tangent spaces is given. Results are applied to the special case of the manifold of minimal projections in \(L(H)\).