  We prove that the projection on a completely entangled subspace S of maximum
dimension in a multipartite quantum system obtained by Parthasarathy is not
positive under partial transpose. We next show that several positive operators
with range in S also have the same property. In this process we construct an
orthonormal basis of S and provide a linking theorem to link the constructions
of completely entangled subspaces due to Parthasarthy, Bhat and Johnston.
