Norms of vector functionals
  We examine the question of when, and how, the norm of a vector functional on an operator algebra can be controlled by the invariant subspace lattice of the algebra. We introduce a related operator algebraic property, and show that it is satisfied by all von Neumann algebras and by all CSL algebras. We exhibit examples of operator algebras that do not satisfy the property or any scaled version of it. 
M. Anoussis, N. Ozawa and I. G. Todorov
math.OA