Projections on von Neumann algebras as limits of elementary operators

If B is a subalgebra of a von Neumann algebra A⊂B(H) and B contains the rank one projections corresponding to an orthonormal basis of H, then a linear B-bimodule projection P on A with range B is of the formP(x)=∑jpjxpjx∈B(H) for orthogonal projections pj in A which are diagonal with respect to the basis. An analogous result holds if A=B(H) and B is a weakly closed ternary ring of operators.