Abstract
This paper characterises the semi-Chebychev subspaces of preduals of von Neumann algebras. As an application it generalises the theorem of Doob, that says that H01 has unique best approximations in L1(T), to a large class of preannihilators of natural non-selfadjoint operator algebras including the nest algebras. Then it studies the semi-Chebychev subspaces of the trace class operators and shows that the only Chebychev *-diagrams are ‘triangular”