  We give a description of a weakly continuous rank preserving map on a
reflexive algebra on complex Hilbert space with commutative completely
distributive subspace lattice. We show that the implementation of a rank
preserving map can be described by the combination of two different types of
maps. We also show that a rank preserving map can be implemented by only one
type if the corresponding lattice is irreducible. We present some examples of
both types of rank preserving map.

Operator Algebras