2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71014The relative position of one subfactor of a factor has been proved quite rich since the work of Jones. We shall show that the theory of relative position of several subspaces of a separable infinite-dimensional Hilbert space is also rich. In finite-dimensonal case, Gelfand and Ponomarev gave a complete classification of indecomposable systems of four subspaces. We construct exotic examples of indecomposable systems of four subspaces in infinite-dimensional Hilbert spaces. We extend their Coxeter functors and defect using Fredholm index. There exist close connections with strongly irreducible operators and transitive lattices.48 pages,improved content for section 12Operator AlgebrasFunctional Analysis46C07 (Primary) 47A15, 16G60 (Secondary)text