On the manifold of complemented principal inner ideals in JB*-triples

The set M of Neher's classes of tripotents in an arbitrary JB*-triple Z is considered and a natural complex-analytic Banach manifold structure is defined on it. The relationship between M and the Grassmann manifold of all complemented principal inner ideals in Z is studied in detail, and the smooth complete vector fields on M are characterized as smooth complete equivariant vector fields on the manifold M of tripotents of Z.