  This paper is mainly devoted to the following question:\ Let $M,N$ be
von~Neumann algebras with $M\subset N$, if there is a completely bounded
projection $P\colon \ N\to M$, is there automatically a contractive projection
$\widetilde P\colon \ N\to M$?
  We give an affirmative answer with the only restriction that $M$ is assumed
semi-finite.
