  We generalize the notion of `diagonal' from the class of CSL algebras to masa
bimodules. We prove that a reflexive masa bimodule decomposes as a sum of two
bimodules, the diagonal and a module generalizing the w*-closure of the
Jacobson radical of a CSL algebra. The latter module turns out to be reflexive,
a result which is new even for CSL algebras. We show that the projection onto
the direct summand contained in the diagonal is contractive and preserves
compactness and reduces rank of operators. Stronger results are obtained when
the module is the reflexive hull of its rank-one subspace.
  Keywords: Operator algebras, reflexivity, TRO, masa-bimodules
