×

Manifolds of tripotents in \(JB^*\)-triples. (English) Zbl 0959.46048

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 \(L(H)\).

MSC:

46L70 Nonassociative selfadjoint operator algebras
46G20 Infinite-dimensional holomorphy
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
17C65 Jordan structures on Banach spaces and algebras
PDFBibTeX XMLCite
Full Text: DOI